Proportions and SimilarityProportions and SimilarityProportions and SimilarityProportions and Similarity
§§ 9.1 Using Ratios and Proportions 9.1 Using Ratios and Proportions
§§ 9.4 Proportional Parts and Triangles 9.4 Proportional Parts and Triangles
§§ 9.3 Similar Triangles 9.3 Similar Triangles
§§ 9.2 Similar Polygons 9.2 Similar Polygons
§§ 9.6 Proportional Parts and Parallel Lines 9.6 Proportional Parts and Parallel Lines
§§ 9.5 Triangles and Parallel Lines 9.5 Triangles and Parallel Lines
§§ 9.7 Perimeters and Similarity 9.7 Perimeters and Similarity
Using Ratios and Proportions Using Ratios and Proportions
You will learn to use ratios and proportions to solve problems.
1) ratio
2) proportion
3) cross products
4) extremes
5) means
Using Ratios and Proportions Using Ratios and Proportions
In 2000, about 180 million tons of solid waste was created in the United States.
The paper made up about 72 million tons of this waste.
The ratio of paper waste to total waste is 72 to 180.
This ratio can be written in the following ways.
72 to 180 72:180 72 ÷ 180180
72
Definition of
Ratio
A ratio is a comparison of two numbers by division.
a to b a:b a ÷ bb
awhere b 0
Using Ratios and Proportions Using Ratios and Proportions
A __________ is an equation that shows two equivalent ratios. proportion
3
2
30
20
Every proportion has two cross products.
In the proportion to the right, the terms 20 and 3 are called the extremes,and the terms 30 and 2 are called the means.
The cross products are 20(3) and 30(2).
The cross products are always _____ in a proportion.
equal
30(2) = 20(3)
60 = 60
Using Ratios and Proportions Using Ratios and Proportions
Theorem 9-1
Property of
Proportions
For any numbers a and c and any nonzero numbers b and d,
, ifd
c
b
a
Likewise, , if bc ad
then , if2
1
10
5 11025
2
1
10
5 then If 11025
bcad then
. thend
c
b
a
Using Ratios and Proportions Using Ratios and Proportions
Solve each proportion:
30
15
2
6
x
15(2x) = 30(6)
30x = 180
x = 6
2
330
x
x
3(x) = (30 – x)2
3x = 60 – 2x
5x = 60
x = 12
Using Ratios and Proportions Using Ratios and Proportions
The gear ratio is the number of teeth on the driving gear to the number of teeth onthe driven gear.
Driving gear
Driven gear
If the gear ratio is 5:2 and the driving gearhas 35 teeth, how many teeth does the driven gear have?
givenratio
equivalentratio=
5
2
35
x=driving gear
driven gear
driving gear
driven gear
5x = 70
x = 14 The driven gear has 14 teeth.
Using Ratios and Proportions Using Ratios and Proportions
Similar Polygons Similar Polygons
You will learn to identify similar polygons.
1) polygons
2) sides
3) similar polygons
4) scale drawing
Similar Polygons Similar Polygons
A polygon is a ______ figure in a plane formed by segments called sides. closed
It is a general term used to describe a geometric figure with at least three sides.
Polygons that are the same shape but not necessarily the same size are called ______________.similar polygons
The symbol ~ is used to show that two figures are similar.
ΔABC ~ ΔDEF
ΔABC is similar to ΔDEFA
BC
D
F E
Similar Polygons Similar Polygons
Definition of
Similar
Polygons
Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are ___________.proportional
CD
A B FE
GH
HE
DA
GH
CD
FG
BC
EF
AB
Polygon ABCD ~ polygon EFGH
HDGC
FBEA
,
,and
Similar Polygons Similar Polygons
Determine if the polygons are similar. Justify your answer.
4
5
4
5 7
6
6
7
1) Are corresponding angles are _________.congruent
2) Are corresponding sides ___________.proportional
4
=
7
6
5
0.66 = 0.71 The polygons are NOT similar!
Similar Polygons Similar Polygons
Find the values of x and y if ΔRST ~ ΔJKL
R
T SJ
L K
4 5
6
7x
y + 2
4=
7
5
x
Write the proportion thatcan be solved for x.
4x = 35
4
38x
4=
7
6
y + 2
Write the proportion thatcan be solved for y.
4(y + 2) = 42
2
18y
4y + 8 = 42
4y = 34
width length
Similar Polygons Similar Polygons
Scale drawings are often used to represent something that is too large or too small to be drawn at actual size.
Contractors use scale drawings to represent the floorplan of a house.
DiningRoom
Kitchen
LivingRoom
Garage
UtilityRoom
1.25 in.
.75 in.
1 in. 1.25 in. .5 in.
Scale: 1 in. = 16 ft.
Use proportions to find the actualdimensions of the kitchen.
1 in=
16 ft
1.25 in.
w ft.
(16)(1.25) = w
20 = w
width is 20 ft.
1 in=
16 ft
.75 in.
L ft.
(16)(.75) = L
12 = L
length is 12 ft.
Similar Polygons Similar Polygons
Similar Triangles Similar Triangles
You will learn to use AA, SSS, and SAS similarity tests fortriangles.
Nothing New!
Similar Triangles Similar Triangles
The Bank of China building in Hong Kong is one of the ten tallest buildings inthe world. Designed by American architect I.M. Pei, the outside of the 70-story buildingis sectioned into triangles which are meant to resemble the trunk of a bambooplant.
Some of the triangles are similar, as shown below.
Similar Triangles Similar Triangles
In previous lessons, you learned several basic tests for determining whethertwo triangles are congruent. Recall that each congruence test involves onlythree corresponding parts of each triangle.
Likewise, there are tests for similarity that will not involve all the parts ofeach triangle.
Postulate
9-1
AA Similarity
If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are ______.similar
C
A B
F
ED
If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF
Similar Triangles Similar Triangles
Theorem 9-2
SSS Similarity
If the measures of the sides of a triangle are ___________
to the measures of the corresponding sides of another triangle, then the triangles are similar.
Two other tests are used to determine whether two triangles are similar.
proportional
C
A B
F
ED
12 3
6
48
1
2
4
8 If
3
6 then the triangles are similarDF
AC
DE
AB If
FE
CB then ΔABC ~ ΔDEF
Similar Triangles Similar Triangles
Theorem 9-3
SAS Similarity
If the measures of two sides of a triangle are ___________
to the measures of two corresponding sides of another triangle and their included angles are congruent, then the triangles are similar.
proportional
C
A B
F
ED
12
48
4
8 If D A and
then ΔABC ~ ΔDEF
DF
AC
DE
AB If
1
2
Similar Triangles Similar Triangles
Determine whether the triangles are similar. If so, tell which similarity testis used and complete the statement.
G
H
K
M
P
J
6
14
10
9
15
21
15=
9
6 10 , the triangles are similar by SSS similarity.Since =21
14
Therefore, ΔGHK ~ Δ JMP
Similar Triangles Similar Triangles
Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet longat the same time that his shadow is 4 feet long.
If Fransisco is 6 feet tall, how tall is the tree?
1) The sun’s rays form congruent angles with the ground.
2) Both Fransisco and the tree form right angles with the ground.
6 ft.
4 ft. 18 ft.
4=
18
6
t
4t = 108
t = 27
The tree is 27 feet tall!
Similar Triangles Similar Triangles
To find the distance across Muddy Pond, he forms similar triangles and measures distances as shown.
Slade is a surveyor.
8 m
10 m
45 m
xWhat is the distanceacross Muddy Pond?
x=
45
10 8
10x = 360
x = 36
It is 36 meters across Muddy Pond!
Similar Triangles Similar Triangles
Proportional Parts and Triangles Proportional Parts and Triangles
You will learn to identify and use the relationships between proportional parts of triangles.
Nothing New!
Proportional Parts and Triangles Proportional Parts and Triangles
In ΔPQR,
TS
RQ
P
Are ΔPQR and ΔPST, similar?
QRST || and intersects the other two sides of ΔPQR.ST
PST PQR corresponding angles
P P
ΔPQR ~ ΔPST. Why? (What theorem / postulate?)
AA Similarity (Postulate 9-1)
Proportional Parts and Triangles Proportional Parts and Triangles
Theorem 9-4
If a line is _______ to one side of a triangle, and intersects the other two sides, then the triangle formed is _______ to the original triangle.
similarparallel
A
B C
D E
then ,|| DEBCIf ΔABC ~ ΔADE.
Proportional Parts and Triangles Proportional Parts and Triangles
TWS
V
R
Complete the proportion:
?
SR
SW
ST
Since RTVW || , ΔSVW ~ ΔSRT.
SR
SW
STSV
Proportional Parts and Triangles Proportional Parts and Triangles
Theorem 9-5
If a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of
__________________.proportional lengths
ED
CB
A
CE
AC
BD
ABDEBC then ,|| If
Proportional Parts and Triangles Proportional Parts and Triangles
CB
HG
A
5
3
x + 5x
4
x
BCGH
of value the Find
.|| figure, the In
5
43
xx
534 xx
1534 xx
15x
Proportional Parts and Triangles Proportional Parts and Triangles
10 ft
6 ft
4 ftBrace
Jacob is a carpenter.
Needing to reinforce this roof rafter, he mustfind the length of the brace.
4=
10
4 x
10x = 16
x = 1 3
5ft
4 ft
x
Proportional Parts and Triangles Proportional Parts and Triangles
Triangles and Parallel Lines Triangles and Parallel Lines
You will learn to use proportions to determine whether linesare parallel to sides of triangles.
Nothing New!
Triangles and Parallel Lines Triangles and Parallel Lines
You know that if a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths (Theorem 9-5).
The converse of this theorem is also true.
Theorem 9-6
If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.
ED
CB
A
9
6
6
4
then , If9
6
6
4 DEBC ||
Triangles and Parallel Lines Triangles and Parallel Lines
Theorem 9-7
If a segment joins the midpoints of two sides of a triangle, then it is parallel to the third side, and its measure equals ________ the measure of the third side.one-half
ED
CB
A
and , of midpoint the is D If AB
2x
x
then , of midpoint the is E AC
and ,|| BCDE BCDE2
1
Triangles and Parallel Lines Triangles and Parallel Lines
ED
CB
A
and , of midpoint the is D If AB
22
11
then , of midpoint the is E AC
and ,|| BCDE BCDE2
1
5
5
8
8
x x 22
2
1
x 11
Use theorem 9 – 7 to find the length of segment DE.
Triangles and Parallel Lines Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ΔMNP.
Complete each statement.
C
BA
N P
M
2) If BC = 14, then MN = ____
1) MP || ____AC
28
3) If mMNP = s, then mBCP = ___s
4) If MP = 18x, then AC = __9x
Triangles and Parallel Lines Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ΔDEF.
2) Find the perimeter of ΔABC
1) Find DE, EF, and FD.14; 10; 16
20
4) Find the ratio of the perimeter of ΔABC to the perimeter of ΔDEF.
20:40 =
C
BA
D F
E
8
753) Find the perimeter of ΔDEF 40
1:2
Triangles and Parallel Lines Triangles and Parallel Lines
A
D
C
B
ABCD is a quadrilateral.
E is the midpoint of AD
F is the midpoint of DC
H is the midpoint of CB
G is the midpoint of BA
Q1) What can you say about EF and GH ?
E
G
H
F
(Hint: Draw diagonal AC .)
They are parallel
Q2) What kind of figure is EFHG ? Parallelogram
Triangles and Parallel Lines Triangles and Parallel Lines
Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines
You will learn to identify and use the relationships betweenparallel lines and proportional parts.
Nothing New!
Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines
On your given paper, draw two (transversals) lines intersecting the parallel lines.
C
D
E
F
A
BLabel the intersections of the transversals and the parallel lines,as shown here.
Measure AB, BC, DE, and EF.
,Calculate each set of ratios:BC
AB
EF
DE
AC
AB
DF
DE,
Do the parallel lines divide the transversals proportionally? Yes
Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines
Theorem 9-8
If three or more parallel lines intersect two transversals,the lines divide the transversals proportionally.
l
m
n
B
A
C F
E
D
If l || m || n
AC
BC
DF
EF=Then
BC
AB
EF
DE= ,
AC
AB
DF
DE= and,
Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines
a
b
c
H
G
J W
V
U
18
12
x
15
Find the value of x.
HJ
GH
VW
UV=
18
12
x
15=
12x = 18(15)
12x = 270
x = 222
1
Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines
Theorem 9-9
If three or more parallel lines cut off congruent segments onone transversal, then they cut off congruent segments onevery transversal.
l
m
n
B
A
C F
E
D
If l || m || n and
Then
AB BC,
DE EF.
Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines
(x +3)
10
(2x – 2)
10
Find the value of x.
F
(x + 3) = (2x – 2)
x + 3 = 2x – 2
5 = x
Theorem 9 - 9
8 8
ED
C
B
A
DE EF
Since AB BC,
Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines
Perimeters and Similarity Perimeters and Similarity
You will learn to identify and use proportional relationships ofsimilar triangles.
1) Scale Factor
Perimeters and Similarity Perimeters and Similarity
These right triangles are similar! Therefore, the measures of their corresponding sides are ___________.
Is there a relationship between the measures of the perimeters of the twotriangles?
8
6 10
12
159
proportional
We know that6
9
8
12
10
15= =
Use the ____________ theoremto calculate the length of thehypotenuse.
Pythagorean
222 bac
2
3=
perimeter of small Δ
perimeter of large Δ=
9 + 12 + 15
6 + 8 + 10=
36
24=
32
Perimeters and Similarity Perimeters and Similarity
perimeter of ΔABCperimeter of ΔDEF
=DEAB
=
Theorem
9-10
If two triangles are similar, then
A
C B
F E
D
the measures of the corresponding perimeters are proportional to the measures of the corresponding sides.
If ΔABC ~ ΔDEF, then
EFBC
=FDCA
Perimeters and Similarity Perimeters and Similarity
27 = 13.5x
The perimeter of ΔRST is 9 units, and ΔRST ~ ΔMNP.
Find the value of each variable. M
N
4.5
P
R
S
T 3
6
z
Yx
perimeter of ΔMNP
perimeter of ΔRST
RSMN
=
13.59x
3=
Theorem 9-10
The perimeter of ΔMNP is 3 + 6 + 4.5
Cross Products
x = 2
RSMN
=STNP
RSMN
=TRPM
23
= y6
23
= z4.5
3y = 12 3z = 9
y = 4 z = 3
Perimeters and Similarity Perimeters and Similarity
DEAB
=EFBC
FDCA
=
Each ratio is equivalent to 21
If ΔABC ~ ΔDEF, then
The ratio found by comparing the measures of corresponding sides of similar triangles is called the constant of proportionality or the ___________.scale factor
A
B C7
53
D
E F14
106
or63
=147
=105
The scale factor of ΔABC to ΔDEF is21
The scale factor of ΔDEF to ΔABC is12
Perimeters and Similarity Perimeters and Similarity
Proportional Parts and Triangles Proportional Parts and Triangles
A
E
D
C
B
I
H
G
F
Step 1) On a piece of lined paper, pick a point on one of the lines and label it A.
Use a straightedge and protractor to draw A so that mA < 90 and only the vertex lies on the line.
Step 2) Extend one side of A down four lines. Label this point E.
Do the same for the other side of A. Label this point I.Now connect points E and I to form ΔAEI.
Step 3) Label the points where the horizontal lines intersect segment AG (B through D).
Label the points where the horizontal lines intersect segment AI (F through H).
AIAHAGAEADAC and , , , , , Measure
GI
AG
CE
AC and ,
:ratios following the compare and Calculate
AI
AH
AE
AD and ,
What can you conclude about the lines through the sides of ΔAEI andparallel to segment EI?
This activity suggests Theorem 9-5.