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8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 159
TERMINOLOGY
4 Geometry 1
Altitude Height Any line segment from a vertex to theopposite side of a polygon that is perpendicular to that side
Congruent triangles Identical triangles that are the sameshape and size Corresponding sides and angles areequal The symbol is
Interval Part of a line including the endpoints
Median A line segment that joins a vertex to theopposite side of a triangle that bisects that side
Perpendicular A line that is at right angles to anotherline The symbol is =
Polygon General term for a many sided plane 1047297gure Aclosed plane (two dimensional) 1047297gure with straight sides
Quadrilateral A four-sided closed 1047297gure such as a squarerectangle trapezium etc
Similar triangles Triangles that are the same shape butdifferent sizes The symbol is yz
Vertex The point where three planes meet The corner ofa 1047297gure
Vertically opposite angles Angles that are formedopposite each other when two lines intersect
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INTRODUCTION
GEOMETRY IS USED IN many areas including surveying building and graphics
These fields all require a knowledge of angles parallel lines and so on and
how to measure them In this chapter you will study angles parallel linestriangles types of quadrilaterals and general polygons
Many exercises in this chapter on geometry need you to prove something
or give reasons for your answers The solutions to geometry proofs only give
one method but other methods are also acceptable
DID YOU KNOW
Geometry means measurement of the earth and comes from Greek Geometry was used in ancient
civilisations such as Babylonia However it was the Greeks who formalised the study of geometry
in the period between 500 BC and AD 300
Notation
In order to show reasons for exercises you must know how to name figures
correctly
bull B
The point is called B
The interval (part of a line) is called AB or BA
If AB and CD are parallel lines we write AB CDlt
This angle is named BAC+ or CAB+ It can sometimes be named A+
Angles can also be written as BAC^
or BAC
This triangle is named ABC3
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This quadrilateral is called ABCD
Line AB is produced to C
DB bisects ABC+
AM is a median of ABCD
AP is an altitude of ABCD
Types of Angles
Acute angle
0 90xc c c1 1
To name a quadrilateral
go around it for example
BCDA is correct but ACBDis not
Producing a line is the same
as extending it
ABD+ and DBC + are
equal
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143Chapter 4 Geometry 1
Right angle
A right angle is 90c
Complementary angles are angles whose sum is 90c
Obtuse angle
x90 180c c c1 1
Straight angle
A straight angle is 180c
Supplementary angles are angles whose sum is 180c
Re1047298ex angle
x180 360c c c1 1
Angle of revolution
An angle of revolution is 360c
Vertically opposite angles
AEC+ and DEB+ are called vertically opposite angles AED+ and CEB+ are
also vertically opposite angles
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Proof
( )
( ) ( )
( )
AEC x
AED x CED
DEB x AEB
x
CEB x CED
AEC DEB AED CEB
180 180
180 180 180
180 180
Let
Then straight angle
Now straight angle
Also straight angle
and`
c
c c c
c c c c
c
c c c
+
+ +
+ +
+ +
+ + + +
=
= -
= - -
=
= -
= =
EXAMPLES
Find the values of all pronumerals giving reasons
1
Solution
( )x ABC
x
x
154 180 180
154 180
26
154 154
is a straight angle
`
c++ =
+ =
=
- -
2
Solution( )x
x
x
x
x
x
2 142 90 360 360
2 232 360
2 232 360
2 128
2 128
64
232 232
2 2
angle of revolution c+ + =
+ =
+ =
=
=
=
- -
Vertically opposite angles are equal
That is AEC DEB+ += and AED CEB+ +=
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145Chapter 4 Geometry 1
3
Solution
( ) y y
y
y
y
y
y
2 30 90 90
3 30 90
3 30 90
3 60
3 60
20
30 30
3 3
right angle c+ + =
+ =
+ =
=
=
=
- -
4
Solution
(
( )
(
x WZX YZV
x
x
y XZY
w WZY XZV
50 165
50 165
115
180 165 180
15
15
50 50
and vertically opposite)
straight angle
and vertically opposite)
c
+ +
+
+ +
+ =
+ =
=
= -
=
=
- -
5
CONTINUED
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Solution
( )
( )
( )
( )
a
b
b
b
b
d
c
90
53 90 180 180
143 180
143 180
37
37
53
143 143
vertically opposite angles
straight angle
vertically opposite angles
similarly
c
=
+ + =
+ =
+ =
=
=
=
- -
6 Find the supplement of 57 12c l
Solution
Supplementary angles add up to 180c
So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l
7 Prove that AB and CD are straight lines
Solution
x x x x
x
x
x
x
6 10 30 5 30 2 10 360
14 80 360
14 280
14 280
20
80 80
14 14
angle of revolution+ + + + + + + =
+ =
=
=
=
- -
^ h
( )
( )
AEC
DEB
20 30
50
2 20 10
50
c
c
c
c
+
+
= +
=
= +
=
These are equal vertically opposite angles
AB and CD are straight lines
C
D A
B
E x 10)
( x +
(5 x + 3 )
x + 30)
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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141Chapter 4 Geometry 1
INTRODUCTION
GEOMETRY IS USED IN many areas including surveying building and graphics
These fields all require a knowledge of angles parallel lines and so on and
how to measure them In this chapter you will study angles parallel linestriangles types of quadrilaterals and general polygons
Many exercises in this chapter on geometry need you to prove something
or give reasons for your answers The solutions to geometry proofs only give
one method but other methods are also acceptable
DID YOU KNOW
Geometry means measurement of the earth and comes from Greek Geometry was used in ancient
civilisations such as Babylonia However it was the Greeks who formalised the study of geometry
in the period between 500 BC and AD 300
Notation
In order to show reasons for exercises you must know how to name figures
correctly
bull B
The point is called B
The interval (part of a line) is called AB or BA
If AB and CD are parallel lines we write AB CDlt
This angle is named BAC+ or CAB+ It can sometimes be named A+
Angles can also be written as BAC^
or BAC
This triangle is named ABC3
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This quadrilateral is called ABCD
Line AB is produced to C
DB bisects ABC+
AM is a median of ABCD
AP is an altitude of ABCD
Types of Angles
Acute angle
0 90xc c c1 1
To name a quadrilateral
go around it for example
BCDA is correct but ACBDis not
Producing a line is the same
as extending it
ABD+ and DBC + are
equal
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143Chapter 4 Geometry 1
Right angle
A right angle is 90c
Complementary angles are angles whose sum is 90c
Obtuse angle
x90 180c c c1 1
Straight angle
A straight angle is 180c
Supplementary angles are angles whose sum is 180c
Re1047298ex angle
x180 360c c c1 1
Angle of revolution
An angle of revolution is 360c
Vertically opposite angles
AEC+ and DEB+ are called vertically opposite angles AED+ and CEB+ are
also vertically opposite angles
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Proof
( )
( ) ( )
( )
AEC x
AED x CED
DEB x AEB
x
CEB x CED
AEC DEB AED CEB
180 180
180 180 180
180 180
Let
Then straight angle
Now straight angle
Also straight angle
and`
c
c c c
c c c c
c
c c c
+
+ +
+ +
+ +
+ + + +
=
= -
= - -
=
= -
= =
EXAMPLES
Find the values of all pronumerals giving reasons
1
Solution
( )x ABC
x
x
154 180 180
154 180
26
154 154
is a straight angle
`
c++ =
+ =
=
- -
2
Solution( )x
x
x
x
x
x
2 142 90 360 360
2 232 360
2 232 360
2 128
2 128
64
232 232
2 2
angle of revolution c+ + =
+ =
+ =
=
=
=
- -
Vertically opposite angles are equal
That is AEC DEB+ += and AED CEB+ +=
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145Chapter 4 Geometry 1
3
Solution
( ) y y
y
y
y
y
y
2 30 90 90
3 30 90
3 30 90
3 60
3 60
20
30 30
3 3
right angle c+ + =
+ =
+ =
=
=
=
- -
4
Solution
(
( )
(
x WZX YZV
x
x
y XZY
w WZY XZV
50 165
50 165
115
180 165 180
15
15
50 50
and vertically opposite)
straight angle
and vertically opposite)
c
+ +
+
+ +
+ =
+ =
=
= -
=
=
- -
5
CONTINUED
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Solution
( )
( )
( )
( )
a
b
b
b
b
d
c
90
53 90 180 180
143 180
143 180
37
37
53
143 143
vertically opposite angles
straight angle
vertically opposite angles
similarly
c
=
+ + =
+ =
+ =
=
=
=
- -
6 Find the supplement of 57 12c l
Solution
Supplementary angles add up to 180c
So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l
7 Prove that AB and CD are straight lines
Solution
x x x x
x
x
x
x
6 10 30 5 30 2 10 360
14 80 360
14 280
14 280
20
80 80
14 14
angle of revolution+ + + + + + + =
+ =
=
=
=
- -
^ h
( )
( )
AEC
DEB
20 30
50
2 20 10
50
c
c
c
c
+
+
= +
=
= +
=
These are equal vertically opposite angles
AB and CD are straight lines
C
D A
B
E x 10)
( x +
(5 x + 3 )
x + 30)
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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142 Maths In Focus Mathematics Extension 1 Preliminary Course
This quadrilateral is called ABCD
Line AB is produced to C
DB bisects ABC+
AM is a median of ABCD
AP is an altitude of ABCD
Types of Angles
Acute angle
0 90xc c c1 1
To name a quadrilateral
go around it for example
BCDA is correct but ACBDis not
Producing a line is the same
as extending it
ABD+ and DBC + are
equal
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143Chapter 4 Geometry 1
Right angle
A right angle is 90c
Complementary angles are angles whose sum is 90c
Obtuse angle
x90 180c c c1 1
Straight angle
A straight angle is 180c
Supplementary angles are angles whose sum is 180c
Re1047298ex angle
x180 360c c c1 1
Angle of revolution
An angle of revolution is 360c
Vertically opposite angles
AEC+ and DEB+ are called vertically opposite angles AED+ and CEB+ are
also vertically opposite angles
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Proof
( )
( ) ( )
( )
AEC x
AED x CED
DEB x AEB
x
CEB x CED
AEC DEB AED CEB
180 180
180 180 180
180 180
Let
Then straight angle
Now straight angle
Also straight angle
and`
c
c c c
c c c c
c
c c c
+
+ +
+ +
+ +
+ + + +
=
= -
= - -
=
= -
= =
EXAMPLES
Find the values of all pronumerals giving reasons
1
Solution
( )x ABC
x
x
154 180 180
154 180
26
154 154
is a straight angle
`
c++ =
+ =
=
- -
2
Solution( )x
x
x
x
x
x
2 142 90 360 360
2 232 360
2 232 360
2 128
2 128
64
232 232
2 2
angle of revolution c+ + =
+ =
+ =
=
=
=
- -
Vertically opposite angles are equal
That is AEC DEB+ += and AED CEB+ +=
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145Chapter 4 Geometry 1
3
Solution
( ) y y
y
y
y
y
y
2 30 90 90
3 30 90
3 30 90
3 60
3 60
20
30 30
3 3
right angle c+ + =
+ =
+ =
=
=
=
- -
4
Solution
(
( )
(
x WZX YZV
x
x
y XZY
w WZY XZV
50 165
50 165
115
180 165 180
15
15
50 50
and vertically opposite)
straight angle
and vertically opposite)
c
+ +
+
+ +
+ =
+ =
=
= -
=
=
- -
5
CONTINUED
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Solution
( )
( )
( )
( )
a
b
b
b
b
d
c
90
53 90 180 180
143 180
143 180
37
37
53
143 143
vertically opposite angles
straight angle
vertically opposite angles
similarly
c
=
+ + =
+ =
+ =
=
=
=
- -
6 Find the supplement of 57 12c l
Solution
Supplementary angles add up to 180c
So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l
7 Prove that AB and CD are straight lines
Solution
x x x x
x
x
x
x
6 10 30 5 30 2 10 360
14 80 360
14 280
14 280
20
80 80
14 14
angle of revolution+ + + + + + + =
+ =
=
=
=
- -
^ h
( )
( )
AEC
DEB
20 30
50
2 20 10
50
c
c
c
c
+
+
= +
=
= +
=
These are equal vertically opposite angles
AB and CD are straight lines
C
D A
B
E x 10)
( x +
(5 x + 3 )
x + 30)
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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152 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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143Chapter 4 Geometry 1
Right angle
A right angle is 90c
Complementary angles are angles whose sum is 90c
Obtuse angle
x90 180c c c1 1
Straight angle
A straight angle is 180c
Supplementary angles are angles whose sum is 180c
Re1047298ex angle
x180 360c c c1 1
Angle of revolution
An angle of revolution is 360c
Vertically opposite angles
AEC+ and DEB+ are called vertically opposite angles AED+ and CEB+ are
also vertically opposite angles
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Proof
( )
( ) ( )
( )
AEC x
AED x CED
DEB x AEB
x
CEB x CED
AEC DEB AED CEB
180 180
180 180 180
180 180
Let
Then straight angle
Now straight angle
Also straight angle
and`
c
c c c
c c c c
c
c c c
+
+ +
+ +
+ +
+ + + +
=
= -
= - -
=
= -
= =
EXAMPLES
Find the values of all pronumerals giving reasons
1
Solution
( )x ABC
x
x
154 180 180
154 180
26
154 154
is a straight angle
`
c++ =
+ =
=
- -
2
Solution( )x
x
x
x
x
x
2 142 90 360 360
2 232 360
2 232 360
2 128
2 128
64
232 232
2 2
angle of revolution c+ + =
+ =
+ =
=
=
=
- -
Vertically opposite angles are equal
That is AEC DEB+ += and AED CEB+ +=
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145Chapter 4 Geometry 1
3
Solution
( ) y y
y
y
y
y
y
2 30 90 90
3 30 90
3 30 90
3 60
3 60
20
30 30
3 3
right angle c+ + =
+ =
+ =
=
=
=
- -
4
Solution
(
( )
(
x WZX YZV
x
x
y XZY
w WZY XZV
50 165
50 165
115
180 165 180
15
15
50 50
and vertically opposite)
straight angle
and vertically opposite)
c
+ +
+
+ +
+ =
+ =
=
= -
=
=
- -
5
CONTINUED
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Solution
( )
( )
( )
( )
a
b
b
b
b
d
c
90
53 90 180 180
143 180
143 180
37
37
53
143 143
vertically opposite angles
straight angle
vertically opposite angles
similarly
c
=
+ + =
+ =
+ =
=
=
=
- -
6 Find the supplement of 57 12c l
Solution
Supplementary angles add up to 180c
So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l
7 Prove that AB and CD are straight lines
Solution
x x x x
x
x
x
x
6 10 30 5 30 2 10 360
14 80 360
14 280
14 280
20
80 80
14 14
angle of revolution+ + + + + + + =
+ =
=
=
=
- -
^ h
( )
( )
AEC
DEB
20 30
50
2 20 10
50
c
c
c
c
+
+
= +
=
= +
=
These are equal vertically opposite angles
AB and CD are straight lines
C
D A
B
E x 10)
( x +
(5 x + 3 )
x + 30)
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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152 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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154 Maths In Focus Mathematics Extension 1 Preliminary Course
All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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156 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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158 Maths In Focus Mathematics Extension 1 Preliminary Course
4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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164 Maths In Focus Mathematics Extension 1 Preliminary Course
Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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Proof
( )
( ) ( )
( )
AEC x
AED x CED
DEB x AEB
x
CEB x CED
AEC DEB AED CEB
180 180
180 180 180
180 180
Let
Then straight angle
Now straight angle
Also straight angle
and`
c
c c c
c c c c
c
c c c
+
+ +
+ +
+ +
+ + + +
=
= -
= - -
=
= -
= =
EXAMPLES
Find the values of all pronumerals giving reasons
1
Solution
( )x ABC
x
x
154 180 180
154 180
26
154 154
is a straight angle
`
c++ =
+ =
=
- -
2
Solution( )x
x
x
x
x
x
2 142 90 360 360
2 232 360
2 232 360
2 128
2 128
64
232 232
2 2
angle of revolution c+ + =
+ =
+ =
=
=
=
- -
Vertically opposite angles are equal
That is AEC DEB+ += and AED CEB+ +=
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145Chapter 4 Geometry 1
3
Solution
( ) y y
y
y
y
y
y
2 30 90 90
3 30 90
3 30 90
3 60
3 60
20
30 30
3 3
right angle c+ + =
+ =
+ =
=
=
=
- -
4
Solution
(
( )
(
x WZX YZV
x
x
y XZY
w WZY XZV
50 165
50 165
115
180 165 180
15
15
50 50
and vertically opposite)
straight angle
and vertically opposite)
c
+ +
+
+ +
+ =
+ =
=
= -
=
=
- -
5
CONTINUED
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146 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
( )
( )
( )
( )
a
b
b
b
b
d
c
90
53 90 180 180
143 180
143 180
37
37
53
143 143
vertically opposite angles
straight angle
vertically opposite angles
similarly
c
=
+ + =
+ =
+ =
=
=
=
- -
6 Find the supplement of 57 12c l
Solution
Supplementary angles add up to 180c
So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l
7 Prove that AB and CD are straight lines
Solution
x x x x
x
x
x
x
6 10 30 5 30 2 10 360
14 80 360
14 280
14 280
20
80 80
14 14
angle of revolution+ + + + + + + =
+ =
=
=
=
- -
^ h
( )
( )
AEC
DEB
20 30
50
2 20 10
50
c
c
c
c
+
+
= +
=
= +
=
These are equal vertically opposite angles
AB and CD are straight lines
C
D A
B
E x 10)
( x +
(5 x + 3 )
x + 30)
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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152 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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145Chapter 4 Geometry 1
3
Solution
( ) y y
y
y
y
y
y
2 30 90 90
3 30 90
3 30 90
3 60
3 60
20
30 30
3 3
right angle c+ + =
+ =
+ =
=
=
=
- -
4
Solution
(
( )
(
x WZX YZV
x
x
y XZY
w WZY XZV
50 165
50 165
115
180 165 180
15
15
50 50
and vertically opposite)
straight angle
and vertically opposite)
c
+ +
+
+ +
+ =
+ =
=
= -
=
=
- -
5
CONTINUED
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Solution
( )
( )
( )
( )
a
b
b
b
b
d
c
90
53 90 180 180
143 180
143 180
37
37
53
143 143
vertically opposite angles
straight angle
vertically opposite angles
similarly
c
=
+ + =
+ =
+ =
=
=
=
- -
6 Find the supplement of 57 12c l
Solution
Supplementary angles add up to 180c
So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l
7 Prove that AB and CD are straight lines
Solution
x x x x
x
x
x
x
6 10 30 5 30 2 10 360
14 80 360
14 280
14 280
20
80 80
14 14
angle of revolution+ + + + + + + =
+ =
=
=
=
- -
^ h
( )
( )
AEC
DEB
20 30
50
2 20 10
50
c
c
c
c
+
+
= +
=
= +
=
These are equal vertically opposite angles
AB and CD are straight lines
C
D A
B
E x 10)
( x +
(5 x + 3 )
x + 30)
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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146 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
( )
( )
( )
( )
a
b
b
b
b
d
c
90
53 90 180 180
143 180
143 180
37
37
53
143 143
vertically opposite angles
straight angle
vertically opposite angles
similarly
c
=
+ + =
+ =
+ =
=
=
=
- -
6 Find the supplement of 57 12c l
Solution
Supplementary angles add up to 180c
So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l
7 Prove that AB and CD are straight lines
Solution
x x x x
x
x
x
x
6 10 30 5 30 2 10 360
14 80 360
14 280
14 280
20
80 80
14 14
angle of revolution+ + + + + + + =
+ =
=
=
=
- -
^ h
( )
( )
AEC
DEB
20 30
50
2 20 10
50
c
c
c
c
+
+
= +
=
= +
=
These are equal vertically opposite angles
AB and CD are straight lines
C
D A
B
E x 10)
( x +
(5 x + 3 )
x + 30)
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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172 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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147Chapter 4 Geometry 1
41 Exercises
1 Find values of all pronumerals
giving reasons
yc 133c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Find the supplement of
(a) 59c (b) 107 31c l
(c) 45 12c l
3 Find the complement of
(a) 48c
(b) 34 23c l
(c) 16 57c l
4 Find the (i) complement and
(ii) supplement of
(a) 43c 81c(b)
27c(c)
(d) 55c
(e) 38c
(f) 74 53c l
(g) 42 24c l
(h) 17 39c l
(i) 63 49c l
(j) 51 9c l
5 (a) Evaluate x Find the complement of(b) x
Find the supplement of(c) x
(2 x +30)c
142c
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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6 Find the values of all
pronumerals giving reasons for
each step of your working
(a)
(b)
(c)
(d)
(e)
(f)
7
Prove that AC and DE are straight
lines
8
Prove that CD bisects AFE+
9 Prove that AC is a straight line
A
B
C
D
(110-3 x )c
(3 x + 70)c
10 Show that + AED is a right angle
A B
C
D E
(50- 8 y)c
(5 y- 20)c
(3 y+ 60)c
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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149Chapter 4 Geometry 1
Parallel Lines
When a transversal cuts two lines it forms pairs of angles When the two
lines are parallel these pairs of angles have special properties
Alternate angles
Alternate angles form
a Z shape Can you
1047297nd another set of
alternate angles
Corresponding angles form
an F shape There are 4 pairs
of corresponding angles Can
you 1047297nd them
If the lines are parallel then alternate angles are equal
Corresponding angles
If the lines are parallel then corresponding angles are equal
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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Cointerior angles
Cointerior angles form
a U shape Can you 1047297nd
another pair
If AEF EFD+ +=
then AB CDlt
If BEF DFG+ +=
then AB CDlt
If BEF DFE 180 c+ ++ =
then AB CDlt
If the lines are parallel cointerior angles are supplementary (ie their sum
is 180c )
Tests for parallel lines
If alternate angles are equal then the lines are parallel
If corresponding angles are equal then the lines are parallel
If cointerior angles are supplementary then the lines are parallel
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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154 Maths In Focus Mathematics Extension 1 Preliminary Course
All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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151Chapter 4 Geometry 1
EXAMPLES
1 Find the value of y giving reasons for each step of your working
Solution
( )
55 ( )
AGF FGH
y AGF CFE AB CD
180 125
55
is a straight angle
corresponding angles`
c c
c
c
+ +
+ + lt
= -
=
=
2 Prove EF GH lt
Solution
( )CBF ABC
CBF HCD
180 120
60
60
is a straight angle
`
c c
c
c
+ +
+ +
= -
=
= =
But CBF + and HCD+ are corresponding angles
EF GH ` lt Can you prove this
in a different way
If 2 lines are both parallel to a third line then the 3 lines are parallel to
each other That is if AB CDlt and EF CDlt then AB EF lt
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1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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152 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find values of all pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2 Prove AB CDlt
(a)
(b)
A
B C
D
E 104c
76c
(c)
42 ExercisesThink about the reasons for
each step of your calculations
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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153Chapter 4 Geometry 1
Types of Triangles
Names of triangles
A scalene triangle has no two sides or angles equal
A right (or right-angled) triangle contains a right angle
The side opposite the right angle (the longest side) is called the
hypotenuse
An isosceles triangle has two equal sides
A
B
C
D
E
F
52c
128c
(d) A B
C
D E
F
G
H
138c
115c
23c
(e)
The angles (called the base angles) opposite the equal sides in an
isosceles triangle are equal
An equilateral triangle has three equal sides and angles
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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164 Maths In Focus Mathematics Extension 1 Preliminary Course
Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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All the angles are acute in an acute-angled triangle
An obtuse-angled triangle contains an obtuse angle
Angle sum of a triangle
The sum of the interior angles in any triangle is 180c
that is a b c 180+ + =
Proof
YXZ a XYZ b YZX c Let andc c c+ + += = =
( )
( )
( )
AB YZ
BXZ c BXZ XZY AB YZ
AXY b
YXZ AXY BXZ AXB
a b c
180
180
Draw line
Then alternate angles
similarly
is a straight angle
`
c
c
c
+ + +
+
+ + + +
lt
lt=
=
+ + =
+ + =
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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155Chapter 4 Geometry 1
Exterior angle of a triangle
Class Investigation
Could you prove the base angles in an isosceles triangle are equal1
Can there be more than one obtuse angle in a triangle2
Could you prove that each angle in an equilateral triangle is3 60c
Can a right-angled triangle be an obtuse-angled triangle4
Can you 1047297nd an isosceles triangle with a right angle in it5
The exterior angle in any triangle is equal to the sum of the two opposite
interior angles That is
x y z+ =
Proof
ABC x BAC y ACD z
CE AB
Let and
Draw line
c c c+ + +
lt
= = =
( )
( )
z ACE ECD
ECD x ECD ABC AB CE
ACE y ACE BAC AB CE
z x y
corresponding angles
alternate angles
`
c
c
c
+ +
+ + +
+ + +
lt
lt
= +
=
=
= +
EXAMPLES
Find the values of all pronumerals giving reasons for each step
1
CONTINUED
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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162 Maths In Focus Mathematics Extension 1 Preliminary Course
3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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164 Maths In Focus Mathematics Extension 1 Preliminary Course
Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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172 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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Solution
( )x
x
xx
53 82 180 180
135 180
135 18045
135 135
angle sum of cD+ + =
+ =
+ =
=
- -
2
Solution
( ) A C x base angles of isosceles+ + D= =
( )x x
x
x
x
x
x
48 180 180
2 48 180
2 48 180
2 132
2 132
66
48 48
2 2
angle sum in a cD+ + =
+ =
+ =
=
=
=
- -
3
Solution
) y y
y
35 14135 141
106
35 35(exterior angle of
`
D+ =+ =
=
- -
This example can be done using the interior sum of angles
( )
( )
BCA BCD
y
y
y
y
180 141 180
39
39 35 180 180
74 180
74 180
106
74 74
is a straight angle
angle sum of
`
c c c
c
c
+ +
D
= -
=
+ + =
+ =
+ =
=
- -
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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157Chapter 4 Geometry 1
1 Find the values of all
pronumerals
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
2 Show that each angle in an
equilateral triangle is 60c
3 Find ACB+ in terms of x
43 ExercisesThink of the reasons
for each step of your
calculations
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4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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158 Maths In Focus Mathematics Extension 1 Preliminary Course
4 Prove AB EDlt
5 Show ABCD is isosceles
6 Line CE bisects BCD+ Find the
value of y giving reasons
7 Evaluate all pronumerals giving
reasons for your working
(a)
(b)
(c)
(d)
8 Prove IJLD is equilateral and
JKLD is isosceles
9 In triangle BCD below BC BD= Prove AB ED
A
B
C
D
E
88c
46c
10 Prove that MN QP
P
N
M
O
Q
32c
75c
73c
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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159Chapter 4 Geometry 1
Congruent Triangles
Two triangles are congruent if they are the same shape and size All pairs of
corresponding sides and angles are equal
For example
We write ABC XYZ D D
Tests
To prove that two triangles are congruent we only need to prove that certain
combinations of sides or angles are equal
Two triangles are congruent if
bull SSS all three pairs of corresponding sides are equal
bull SAS two pairs of corresponding sides and their included angles are
equal
bull AAS two pairs of angles and one pair of corresponding sides are equal
bull RHS both have a right angle their hypotenuses are equal and one
other pair of corresponding sides are equal
EXAMPLES
1 Prove that OTS OQP D D where O is the centre of the circle
CONTINUED
The included angle
is the angle between
the 2 sides
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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180 Maths In Focus Mathematics Extension 1 Preliminary Course
Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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Solution
S
A
S
OS OQ
TOS QOP
OT OP
OTS OQP
(equal radii)
(vertically opposite angles)
(equal radii)
by SAS`
+ +
D D
=
=
=
2 Which two triangles are congruent
Solution
To 1047297nd corresponding sides look at each side in relation to the angles
For example one set of corresponding sides is AB DF GH and JL
ABC JKL A(by S S)D D
3 Show that triangles ABC and DEC are congruent Hence prove that
AB ED=
Solution
( )
( )
( )
( )
A
A
S
BAC CDE AB ED
ABC CED
AC CD
ABC DEC
AB ED
alternate angles
similarly
given
by AAS
corresponding sides in congruent s
`
`
+ +
+ +
lt
D D
D
=
=
=
=
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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161Chapter 4 Geometry 1
1 Are these triangles congruent
If they are prove that they are
congruent
(a)
(b)
X
Z
Y
B
C
A
4 7 m
2 3 m
2 3 m
4 7 m 110c 1 1 0
c
(c)
(d)
(e)(e
2 Prove that these triangles are
congruent
(a)
(b)
(c)
(d)
(e)
44 Exercises
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3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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162 Maths In Focus Mathematics Extension 1 Preliminary Course
3 Prove that
(a) ∆ ABD is congruent to ∆ ACD
(b) AB bisects BC given ABCD is
isosceles with AB AC=
4 Prove that triangles ABD and CDB
are congruent Hence prove that
AD BC=
5 In the circle below O is the centre
of the circle
O
A
B
D
C
Prove that(a) OABT and OCDT
are congruent
Show that(b) AB CD=
6 In the kite ABCD AB AD= and
BC DC=
A
B D
C
Prove that(a) ABCT and ADCT
are congruent
Show that(b) ABC ADC+ +=
7 The centre of a circle is O and AC
is perpendicular to OB
O
A
B
C
Show that(a) OABT and OBCT
are congruent
Prove that(b) ABC 90c+ =
8 ABCF is a trapezium with
AF BC= and FE CD= AE and BD
are perpendicular to FC
D
A B
C F E
Show that(a) AFET and BCDT
are congruent
Prove that(b) AFE BCD+ +=
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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164 Maths In Focus Mathematics Extension 1 Preliminary Course
Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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163Chapter 4 Geometry 1
9 The circle below has centre O and
OB bisects chord AC
O
A
B
C
Prove that(a) OABT is congruent
to OBCT
Prove that(b) OB is perpendicular
to AC
10 ABCD is a rectangle as shown
below
D
A B
C
Prove that(a) ADCT is
congruent to BCDT
Show that diagonals(b) AC and
BD are equal
Investigation
The triangle is used in many
structures for example trestle
tables stepladders and roofs
Find out how many different ways
the triangle is used in the building
industry Visit a building site orinterview a carpenter Write a
report on what you 1047297nd
Similar Triangles
Triangles for example ABC and XYZ are similar if they are the same shape but
different sizes
As in the example all three pairs of corresponding angles are equal
All three pairs of corresponding sides are in proportion (in the same ratio)
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Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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164 Maths In Focus Mathematics Extension 1 Preliminary Course
Application
Similar 1047297gures are used in many areas including maps scale drawings models
and enlargements
EXAMPLE
1 Find the values of x and y in similar triangles CBA and XYZ
Solution
First check which sides correspond to one another (by looking at their
relationships to the angles)
YZ and BA XZ and CA and XY and CB are corresponding sides
CA XZ
CB XY
y
y 4 9 3 6
5 4
3 6 4 9 5 4
`
=
=
=
We write XYZ D ABC ltD
XYZ D is three times larger than ABCD
AB XY
AC XZ
BCYZ
AB XY
AC XZ
BCYZ
26
3
412
3
515 3
`
= =
= =
= =
= =
This shows that all 3 pairs
of sides are in proportion
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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165Chapter 4 Geometry 1
y
BAYZ
CB XY
x
x
x
3 6
4 9 5 4
7 35
2 3 3 65 4
3 6 2 3 5 4
3 6
2 3 5 4
3 45
=
=
=
=
=
=
=
Two triangles are similar if
three pairs ofbull corresponding angles are equal
three pairs ofbull corresponding sides are in proportion
two pairs ofbull sides are in proportion and their included angles
are equal
If 2 pairs of angles are
equal then the third
pair must also be equal
EXAMPLES
1Prove that triangles(a) ABC and ADE are similar
Hence 1047297nd the value of(b) y to 1 decimal place
Solution
(a) A+ is common
ADE D
( )( )
( )
ABC ADE BC DE ACB AED
ABC
corresponding anglessimilarly
3 pairs of angles equal`
+ +
+ +
lt
ltD
=
=
(b)
CONTINUED
Tests
There are three tests for similar triangles
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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172 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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166 Maths In Focus Mathematics Extension 1 Preliminary Course
AE
BC DE
AC AE
y
y
y
2 4 1 9
4 3
3 7 2 42 4 3 7 4 3
2 43 7 4 3
6 6
4 3
= +
=
=
=
=
=
=
2 Prove WVZ D XYZ ltD
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
3515
73
146
73
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are
equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles
When two (or more) transversals cut a series of parallel lines the
ratios of their intercepts are equal
AB BC DE EF
BC AB
EF DE
That is
or
=
=
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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180 Maths In Focus Mathematics Extension 1 Preliminary Course
Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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167Chapter 4 Geometry 1
Proof
Draw DG and EH parallel to AC
`
EHF D
`
`
( )
( )
( )
( )
( )
( )
DG AB
EH BC
BC AB
EH DG
GDE HEF DG EH
DEG EFH BE CF
DGE EHF
DGE
EH DG
EF DE
BC AB
EF DE
1
2
Then opposite sides of a parallelogram
Also (similarly)
corresponding s
corresponding s
angle sum of s
So
From (1) and (2)
+ + +
+ + +
+ +
lt
lt
lt
D
D
=
=
=
=
=
=
=
=
EXAMPLES
1 Find the value of x to 3 signi1047297cant 1047297gures
Solution
x
x
x
8 9 9 31 5
9 3 8 9 1 5
9 3
8 9 1 5
1 44
ratios of intercepts on parallel lines
=
=
=
=
^ h
CONTINUED
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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172 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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168 Maths In Focus Mathematics Extension 1 Preliminary Course
2 Evaluate x and y to 1 decimal place
Solution
Use either similar triangles or ratios of intercepts to 1047297nd x You must use
similar triangles to 1047297nd y
x
x
y
y
5 8 3 42 7
3 4
2 7 5 8
4 6
7 1 3 4
2 7 3 4
3 46 1 7 1
12 7
=
=
=
= +
=
=
1 Find the value of all pronumerals
to 1 decimal place where
appropriate
(a)
(b)
(c)
(d)
(e)
45 Exercises
These ratios come
from intercepts on
parallel lines
These ratios come from
similar triangles
Why
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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170 Maths In Focus Mathematics Extension 1 Preliminary Course
9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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172 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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169Chapter 4 Geometry 1
(f)
143
a
4 6 c
1 9 c
1 1 5 c
4 6 c
x c
91
257
89 y
(g)
2 Evaluate a and b to 2 decimal
places
3 Show that ABCD and CDED are
similar
4 EF bisects GFD+ Show that
DEF D
and FGED
are similar
5 Show that ABCD and DEF D are
similar Hence 1047297nd the value of y
42
49
686
13
588182
A
C
B D
E F
yc87c
52c
6 The diagram shows two
concentric circles with centre O
Prove that(a) D OCDOAB ltD
If radius(b) OC 5 9 c m= and
radius OB 8 3 cm= and the
length of CD 3 7 cm= 1047297nd the
length of AB correct to 2 decimal
places
7 (a) Prove that ADED ABC ltD
Find the values of(b) x and y
correct to 2 decimal places
8 ABCD is a parallelogram with
CD produced to E Prove that
CEBD ABF ltD
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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9 Show that ABC D AED ltD Find
the value of m
10 Prove that ABCD and ACDD are
similar Hence evaluate x and y
11 Find the values of all
pronumerals to 1 decimal place
(a)
(b)
(c)
(d)
(e)
12 Show that
(a) BC AB
FG AF
=
(b) AC AB
AG AF
=
(c)CE BD
EG DF
=
13 Evaluate a and b correct to
1 decimal place
14 Find the value of y to 2
signi1047297cant 1047297gures
15 Evaluate x and y correct to
2 decimal places
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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172 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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171Chapter 4 Geometry 1
Pythagorasrsquo Theorem
DID YOU KNOW
The triangle with sides in the
proportion 345 was known to be
right angled as far back as ancient
Egyptian times Egyptian surveyors
used to measure right angles by
stretching out a rope with knots tied
in it at regular intervals
They used the rope for forming
right angles while building and
dividing 1047297elds into rectangular plots
It was Pythagoras (572ndash495 BC)
who actually discovered the
relationship between the sides of the
right-angled triangle He was able to
generalise the rule to all right-angled triangles
Pythagoras was a Greek mathematician
philosopher and mystic He founded the Pythagorean
School where mathematics science and philosophy
were studied The school developed a brotherhood and
performed secret rituals He and his followers believed
that the whole universe was based on numbers
Pythagoras was murdered when he was 77 and the
brotherhood was disbanded
The square on the hypotenuse in any right-angled triangle is equal to the
sum of the squares on the other two sides
c a b
c a b
That is
or
2 2 2
2 2
= +
= +
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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Proof
Draw CD perpendicular to AB
Let AD x DB y = =
Then x y c + =
In ADCD and ABCD
A+ is common
D
D
( ) ABC
ABC
equal corresponding s+
ADC ACB
ADC
AB AC
AC AD
c b
bx
b xc
BDC
BC DB
AB BC
a
y
c a
a yc
a b yc xc
c y x
c c
c
90
Similarly
Now
2
2
2 2
2
`
c+ +
lt
lt
D
D
= =
=
=
=
=
=
=
+ = +
= +
=
=
^]
hg
EXAMPLES
1 Find the value of x correct to 2 decimal places
Solution
c a b
x 7 4
49 16
65
2 2 2
2 2 2
= +
= +
= +
=
c a b ABCIf then must be right angled2 2 2D= +
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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173Chapter 4 Geometry 1
x 65
8 06 to 2 decimal places
=
=
2 Find the exact value of y
Solution
c a b
y
y y
y
8 4
64 1648
48
16 3
4 3
2 2 2
2 2 2
2
2
`
= +
= +
= +
=
=
=
=
3 Find the length of the diagonal in a square with sides 6 cm Answer to
1 decimal place
Solution
6 cm
cm
c a b
c
6 6
72
72
8 5
2 2 2
2 2
= +
= +
=
=
=
So the length of the diagonal is 85 cm
Leave the answer in
surd form for the exact
answer
CONTINUED
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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180 Maths In Focus Mathematics Extension 1 Preliminary Course
Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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174 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumerals
correct to 1 decimal place(a)
(b)
(c)
(d)
2 Find the exact value of all
pronumerals(a)
(b)
(c)
(d)
46 Exercises
4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle
is right angled
Solution
68 cm
85 cm
51 cm
Let c 8 5= (largest side) and a and b the other two smaller sides
a b
c
c a b
5 1 6 8
72 25
8 5
72 25
2 2 2 2
2 2
2 2 2`
+ = +
=
=
=
= +
So the triangle is right angled
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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175Chapter 4 Geometry 1
3 Find the slant height s of a
cone with diameter 68 m and
perpendicular height 52 m to
1 decimal place
4 Find the length of CE correct
to 1 decimal place in this
rectangular pyramid 86 AB cm=
and 159 CF cm=
5 Prove that ABCD is a right-angled
triangle
6 Show that XYZ D is a right-angled
isosceles triangle
X
Y Z 1
12
7 Show that AC BC2=
8 (a) Find the length of diagonal
AC in the 1047297gure
Hence or otherwise prove(b)
that AC is perpendicular to DC
9 Find the length of side AB in
terms of b
10 Find the exact ratio of YZ XY
in
terms of x and y in XYZ D
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11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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176 Maths In Focus Mathematics Extension 1 Preliminary Course
11 Show that the distance squared
between A and B is given by
d t t 13 180 6252 2= - +
12 An 850 mm by 1200 mm gate
is to have a diagonal timber
brace to give it strength To what
length should the timber be cut
to the nearest mm
13 A rectangular park has a length of620 m and a width of 287 m If I
walk diagonally across the park
how far do I walk
14 The triangular garden bed below
is to have a border around it
How many metres of border are
needed to 1 decimal place
15 What is the longest length of
stick that will 1047297t into the box
below to 1 decimal place
16 A ramp is 45 m long and 13 m
high How far along the ground
does the ramp go Answer correct
to one decimal place
45 m
13 m
17 The diagonal of a television
screen is 72 cm If the screen is
58 cm high how wide is it
18 A property has one side 13 km
and another 11 km as shown
with a straight road diagonally
through the middle of the
property If the road is 15 km
long show that the property is
not rectangular
13 km
11 km
15 km
19 Jodie buys a ladder 2 m long and
wants to take it home in the boot
of her car If the boot is 12 m by
07 m will the ladder 1047297t
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
8142019 Geometry 1 i
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
8142019 Geometry 1 i
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180 Maths In Focus Mathematics Extension 1 Preliminary Course
Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
8142019 Geometry 1 i
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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8142019 Geometry 1 i
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
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177Chapter 4 Geometry 1
Types of Quadrilaterals
A quadrilateral is any four-sided 1047297gure
In any quadrilateral the sum of the interior angles is 360c
20 A chord AB in a circle with
centre O and radius 6 cm has a
perpendicular line OC as shown
4 cm long
A
B
O
C
6 cm
4 cm
By 1047297nding the lengths of(a) AC
and BC show that OC bisects the
chord
By proving congruent(b)
triangles show that OC bisects
the chord
Proof
Draw in diagonal AC
180 ( )
( )
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360
360
angle sum of
similarly
That is
`
c
c
c
c
+ + +
+ + +
+ + + + + +
+ + + +
D+ + =
+ + =
+ + + + + =
+ + + =
8142019 Geometry 1 i
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4059
179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
8142019 Geometry 1 i
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180 Maths In Focus Mathematics Extension 1 Preliminary Course
Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
8142019 Geometry 1 i
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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178 Maths In Focus Mathematics Extension 1 Preliminary Course
opposite sidesbull of a parallelogram are equal
bull opposite angles of a parallelogram are equal
bull diagonals in a parallelogram bisect each other
each diagonal bisects the parallelogram into twobull
congruent triangles
A quadrilateral is a parallelogram if
both pairs ofbull opposite sides are equal
both pairs ofbull opposite angles are equal
onebull pair of sides is both equal and parallel
thebull diagonals bisect each other
These properties can
all be proven
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i
Solution
120 56 90 360
266 360
94
angle sum of quadrilaterali
i
i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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180 Maths In Focus Mathematics Extension 1 Preliminary Course
Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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179Chapter 4 Geometry 1
Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram and alsobull
diagonals are equalbull
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate
For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the
Leaning Tower of Pisa
It can be proved that
all sides are equal
If one angle is a right
angle then you can
prove all angles are
right angles
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram and alsobull
diagonals bisect at right anglesbull
diagonals bisect the angles of the rhombusbull
Rectangle
PROPERTIES
PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
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1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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180 Maths In Focus Mathematics Extension 1 Preliminary Course
Square
A square is a rectangle with a pair of adjacent sides equal
bull the same as for rectangle and also
diagonals are perpendicularbull
diagonals make angles ofbull 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if
all sides are equalbull
diagonals bisect each other at right anglesbull
TESTS
PROPERTIES
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
8142019 Geometry 1 i
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
8142019 Geometry 1 i
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
8142019 Geometry 1 i
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
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181Chapter 4 Geometry 1
EXAMPLES
1 Find the values of i x and y giving reasons
Solution
( )
( )
( )
x
y
83
6 7
2 3
opposite s in gram
cm opposite sides in gram
cm opposite sides in gram
c + lt
lt
lt
i =
=
=
2 Find the length of AB in square ABCD as a surd in its simplest form if
6 BD cm=
Solution
( )
( )
AB x
ABCD AB AD x
A 90
Let
Since is a square adjacent sides equal
Also by definitionc+
=
= =
=
By Pythagorasrsquo theorem
3
c a b
x x
x
x
x
6
36 2
18
182 cm
2 2 2
2 2 2
2
2
`
= +
= +
=
=
=
=
CONTINUED
8142019 Geometry 1 i
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
8142019 Geometry 1 i
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
8142019 Geometry 1 i
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
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8142019 Geometry 1 i
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
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182 Maths In Focus Mathematics Extension 1 Preliminary Course
1 Find the value of all pronumeralsgiving reasons
(a)
(b)
(c)
(d)
(e)
(f)
(g)
47 Exercises
3
Two equal circles have centres(a) O and P respectively Prove that OAPB
is a rhombus
Hence or otherwise show that(b) AB is the perpendicular bisector
of OP
Solution
(a) ( )
( )
OA OB
PA PBOA OB PA PB
equal radii
similarlySince the circles are equal
=
=
= = =
` since all sides are equal OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors(b)
Since OAPB is a rhombus with diagonals AB and OP AB is the
perpendicular bisector of OP
8142019 Geometry 1 i
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
8142019 Geometry 1 i
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
8142019 Geometry 1 i
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
8142019 Geometry 1 i
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
8142019 Geometry 1 i
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
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8142019 Geometry 1 i
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
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183Chapter 4 Geometry 1
2 Given AB AE= prove CD is
perpendicular to AD
3 (a) Show that C xc+ = and
( ) B D x180 c+ += = -
Hence show that the sum of(b)
angles of ABCD is 360c
4 Find the value of a and b
5 Find the values of all
pronumerals giving reasons
(a)
(b)
(c)
(d)
(e)
7
y
3 x
x + 6
(f)
6 In the 1047297gure BD bisects
ADC+ Prove BD also bisects
ABC+
7 Prove that each 1047297gure is a
parallelogram
(a)
(b)
8142019 Geometry 1 i
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4659
185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
8142019 Geometry 1 i
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
8142019 Geometry 1 i
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5059
189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
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8142019 Geometry 1 i
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
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184 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
8 Evaluate all pronumerals
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9 The diagonals of a rhombus
are 8 cm and 10 cm long Find
the length of the sides of the
rhombus
10 ABCD is a rectangle with
EBC 59c+ = Find ECB EDC+ +
and ADE+
11 The diagonals of a square are
8 cm long Find the exact lengthof the side of the square
12 In the rhombus ECB 33c+ =
Find the value of x and y
Polygons
A polygon is a closed plane 1047297gure with straight sides
A regular polygon has all sides and all interior angles equal
8142019 Geometry 1 i
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185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
8142019 Geometry 1 i
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
8142019 Geometry 1 i
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5059
189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5259
191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4659
185Chapter 4 Geometry 1
Proof
Draw any n -sided polygon and divide it into n triangles as
shown Then the total sum of angles is n 180 c or 180 n
But this sum includes all the angles at O So the sum of
interior angles is 180 360 n c-
That is S n
n
180 360
2 180 c
= -
= -] g
EXAMPLES
4-sided(square)
3-sided(equilateral
triangle)
5-sided(pentagon)
6-sided(hexagon)
8-sided(octagon)
10-sided(decagon)
DID YOU KNOW
Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When
he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and
compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including
correctly calculating where the magnetic south pole is and designing a lens to correct
astigmatism
He was the director of the Goumlttingen Observatory for 40 years It is said that he did not
become a professor of mathematics because he did not like teaching
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or c
= -
= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon Then the sum of both the
exterior and interior angles is n 180 c
n
n n
n n
180
180 180 360
180 180 360
360
Sum of exterior angles sum of interior angles c
c
c
c
= -
= - -
= - +
=
] g
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4759
186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4859
187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4959
188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5059
189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
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191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
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193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
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195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
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8142019 Geometry 1 i
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
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186 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1 Find the sum of the interior angles of a regular polygon with 15 sides
How large is each angle
Solution
( )
( )
n
S n
15
0
15 0
0
2340
2 18
2 18
13 18
c
c
c
c
=
= -
= -
=
=
Each angle has size 2340 15 156c c=
2 Find the number of sides in a regular polygon whose interior angles
are 140c
Solution
Let n be the number of sides
Then the sum of interior angles is 140n
( )
( )
S n
n n
n
n
n
2 180
140 2 180
180 360
360 40
9
But
So
c
c
= -
= -
= -
=
=
So the polygon has 9 sides
There are n sides and so n
angles each 140 c
1 Find the sum of the interior
angles of
a pentagon(a)
a hexagon(b)
an octagon(c)a decagon(d)
a 12-sided polygon(e)
an 18-sided polygon(f)
2 Find the size of each interior
angle of a regular
pentagon(a)
octagon(b)
12-sided polygon(c)
20-sided polygon(d)
15-sided polygon(e)
3 Find the size of each exterior
angle of a regular
hexagon(a)
decagon(b)
octagon(c)15-sided polygon(d)
4 Calculate the size of each
interior angle in a regular 7-sided
polygon to the nearest minute
5 The sum of the interior angles of
a regular polygon is 1980c
How many sides has the(a)
polygon
Find the size of each interior(b)
angle to the nearest minute
48 Exercises
8142019 Geometry 1 i
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187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5059
189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5259
191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
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8142019 Geometry 1 i
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198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4859
187Chapter 4 Geometry 1
6 Find the number of sides of a
regular polygon whose interior
angles are 157 30c l
7 Find the sum of the interior
angles of a regular polygon whose
exterior angles are 18c
8 A regular polygon has interior
angles of 156c Find the sum of its
interior angles
9 Find the size of each interior
angle in a regular polygon if
the sum of the interior angles is
5220c
10 Show that there is no regular
polygon with interior angles of
145c
11 Find the number of sides of a
regular polygon with exterior
angles
(a) 40c
(b) 03 c
(c) 45c
(d) 36c
(e) 12c
12 ABCDEF is a regular hexagon
F
E D
A B
C
Show that triangles(a) AFE and
BCD are congruent
Show that(b) AE and BD are
parallel
13 A regular octagon has a
quadrilateral ACEG inscribed as
shown
D
A
B
E
C
F
G
H
Show that ACEG is a square
14 In the regular pentagon below
show that EAC is an isosceles
triangle
D
A
B E
C
15 (a) Find the size of each exterior
angle in a regular polygon with
side p
Hence show that each interior(b)
angle is
( )
p
p180 2-
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 4959
188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5059
189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5259
191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5359
192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
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188 Maths In Focus Mathematics Extension 1 Preliminary Course
Areas
Most areas of plane 1047297gures come from the area of a rectangle
Rectangle
A lb=
Square
A x2=
Triangle
A bh21
=
Proof
h
b
Draw rectangle ABCD where b length= and h breadth=
A square is a
special rectangle
The area of a triangle
is half the area of a
rectangle
8142019 Geometry 1 i
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189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5259
191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5059
189Chapter 4 Geometry 1
bharea
21
21
21
21
` =
DEF AEFD CEF EBCF Area area and area areaD D= =
CDE ABCDarea` D =
A bhThat is =
area
A bh=
Proof
In parallelogram ABCD produce DC to E and draw BE perpendicular to CE
Then ABEF is a rectangle
Area ABEF bh=
In ADF D and BCED
( )
( )
AFD BEC
AF BE h
AD BC
ADF BCE
ADF BCE
ABCD ABEF
bh
90
opposite sides of a rectangle
opposite sides of a parallelogram
by RHS
area area
So area area
`
`
c+ +
D D
D D
= =
= =
=
=
=
=
Rhombus
The area of a
parallelogram is the
same as the area of
two triangles
A xy 21
=
(x and y are lengths of diagonals)
Parallelogram
8142019 Geometry 1 i
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190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5259
191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5359
192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
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196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5159
190 Maths In Focus Mathematics Extension 1 Preliminary Course
( ) A h a b21
= +
Proof
DE x
DF x a
FC b x a
b x a
Let
Then
`
=
= +
= - +
= - -
] g
Proof
Let AC x= and BD y =
By properties of a rhombus
AE EC x21
= = and DE EB y 21
= =
Also AEB 90c+ =
ABC x y
xy
ADC x y
xy
xy xy
xy
Area
Area
total area of rhombus
21
21
41
21
21
41
41
41
21
`
D
D
=
=
=
=
= +
=
Trapezium
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5259
191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5359
192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5559
194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5759
196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5259
191Chapter 4 Geometry 1
A r 2r=
EXAMPLES
1 Find the area of this trapezium
Solution
( )
( ) ( )
24
A h a b
4 7 5
2 12
m2
21
21
= +
= +
=
=
2 Find the area of the shaded region in this 1047297gure
8 c
m
7 c m
21 cm
42 cm
CONTINUED
( )
( )
( )
ADE ABFE BFC
xh ah b x a h
h x a b x a
h a b
2
Area trapezium area area rectangle area
21
21
21
2
1
D D= + +
= + + - -
= + + - -
= +
Circle
You will study the circle in
more detail in Chapter 9
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5359
192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5759
196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
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192 Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
lb
lb
8 9 12 1
107 69
3 7 4 2
15 54
107 69 15 54
92 15
Area large rectangle
cm
Area small rectangle
cm
shaded area
cm
2
2
2
`
=
=
=
=
=
=
= -
=
3 A park with straight sides of length 126 m and width 54 m has semi-
circular ends as shown Find its area correct to 2 decimal places
2 m
5 4 m
Solution
-Area of 2 semicircles area of 1 circle=
2
( )
r
A r
254
27
27
2290 22 m
2
2
r
r
=
=
=
=
=
126 54
6804
2290 22 6804
9094 22
Area rectangle
Total area
m2
=
=
= +
=
1 Find the area of each 1047297gure
(a)
(b)
49 Exercises
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5759
196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5459
193Chapter 4 Geometry 1
(c)
(d)
(e)
(f)
(g)
2 Find the area of a rhombus with
diagonals 23 m and 42 m
3 Find each shaded area(a)
(b)
(c)
(d)
(e)
6 c m
2 cm
4 Find the area of each 1047297gure
(a)
(b)
8142019 Geometry 1 i
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194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5759
196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5559
194 Maths In Focus Mathematics Extension 1 Preliminary Course
(c)
(d)
(e)
5 Find the exact area of the 1047297gure
6 Find the area of this 1047297gure
correct to 4 signi1047297cant 1047297gures
The arch is a semicircle
7 Jenny buys tiles for the 1047298oor of
her bathroom (shown top next
column) at $4550 per m2 How
much do they cost altogether
8 The dimensions of a battleaxe
block of land are shown below
Find its area(a)
A house in the district where(b)
this land is can only take up 55
of the land How large (to the
nearest m2 ) can the area of the
house beIf the house is to be a(c)
rectangular shape with width
85 m what will its length be
9 A rhombus has one diagonal
25 cm long and its area is
600 cm2 Find the length of
its other diagonal and(a)
its side to the nearest cm(b)
10 The width w of a rectangle is
a quarter the size of its length
If the width is increased by 3units while the length remains
constant 1047297nd the amount of
increase in its area in terms of w
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5759
196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5659
195Chapter 4 Geometry 1
Test Yourself 4
The perimeter
is the distance
around the outs
of the 1047297gure
1 Find the values of all pronumerals
(a)
(b)
(c)
x (d)
O is the centre
of the circle)
(e)
(f)
(g)
2 Prove that AB and CD are parallel lines
3 Find the area of the 1047297gure to 2 decimalplaces
4 (a) Prove that triangles ABC and ADE are
similar
Evaluate(b) x and y to 1 decimal place
5 Find the size of each interior angle in a
regular 20-sided polygon
6 Find the volume of a cylinder with radius
57 cm and height 10 cm correct to
1 decimal place
7 Find the perimeter of the triangle below
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5759
196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5759
196 Maths In Focus Mathematics Extension 1 Preliminary Course
8 (a) Prove triangles ABC and ADC are
congruent in the kite below
Prove triangle(b) AOB and COD are
congruent (O is the centre of the circle)
9 Find the area of the 1047297gure below
10 Prove triangle ABC is right angled
11 Prove AG AF
AC AB
=
12 Triangle ABC is isosceles and AD bisects
BC
Prove triangles(a) ABD and ACD are
congruent
Prove(b) AD and BC are perpendicular
13 Triangle ABC is isosceles with AB AC=
Show that triangle ACD is isosceles
14 Prove that opposite sides in any
parallelogram are equal
15 A rhombus has diagonals 6 cm and 8 cm
Find the area of the rhombus(a)
Find the length of its side(b)
16 The interior angles in a regular polygon
are 140c How many sides has the
polygon
17 Prove AB and CD are parallel
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5859
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long
8142019 Geometry 1 i
httpslidepdfcomreaderfullgeometry-1-i 5959
198 Maths In Focus Mathematics Extension 1 Preliminary Course
7 Prove that the diagonals in a square
make angles of 45c with the sides
8 Prove that the diagonals in a kite are
perpendicular
9 Prove that MN is parallel to XY
10 Evaluate x
11 The letter Z is painted on a billboard
Find the area of the letter(a)
Find the exact perimeter of the letter(b)
12 Find the values of x and y correct to
1 decimal place
13 Find the values of x and y correct to
2 decimal places
14 ABCD is a square and BD is produced to
E such that DE BD21
=
Show that(a) ABCE is a kite
Prove that(b) DE x
2
2= units when
sides of the square are x units long