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Curriculum Ready
Pythagoras Theorem
PythagorasTheorem
7ISERIES TOPIC
1Pythagoras’ TheoremMathletics Passport © 3P Learning
The numbers 3, 4, 5 have the following relationship:
3 4 5
9 16 25
2 2 2+ =
+ =
Find another group of three whole numbers that includes the number 14 and has the same relationship.psst! the other two numbers are somewhere between 45 and 55!
Work through the book for a great way to do this
Give this a go!
Fill in these spaces with any other interesting facts you can find about Pythagoras.
One of his most recognised discoveries was the relationship between the side lengths of all right-angled triangles.
In the world of Mathematics, Pythagoras is a legend. He lived from 580 BC – 500 BC.
2 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
Pythagoras’ TheoremHow does it work?
Right-angled triangles
These special triangles all have a right-angle (angle of size 90o) as one of the internal angles.
For each of these right-angled triangles, name the hypotenuse and then draw in the right-angle.
Short side
Hypotenuse(Longest side)
Other short side
opposite
(i) (ii)
The hypotenuse is the longest side
` hypotenuse = side c
The right-angle is the angle opposite the hypotenuse
The hypotenuse is the longest side
` hypotenuse = side XZ
The hypotenuse is always opposite the right-angle
Z
Y
Xa
c
b
Z
Y
X
Sides are lower case and corners are CAPITALS
opposite
hypotenuse
The two shorter sides are always perpendicular to each other.
Perpendicular = 90o
opposite
hypotenuse
a
c
b
Right angle
7ISERIES TOPIC
3Pythagoras’ TheoremMathletics Passport © 3P Learning
How does it work? Your Turn Pythagoras’ Theorem
Right-angled triangles
1 For each of these right-angled triangles, name the hypotenuse and draw the right-angle in the correct position.
2
M
NL
a b
Hypotenuse is side:
Hypotenuse is side:
Hypotenuse is side:
Hypotenuse is side:
a b
c d
y
Q
RP
D
F
E
xz
k
jl
ca
b
Hypotenuse is side: Hypotenuse is side:
Name the hypotenuse for each of these badly drawn triangles:
RIGHT-ANGLED
TRIANG
LES
RIGHT-ANGLED TRIANGLES..../...../20...
4 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
How does it work? Pythagoras’ Theorem
Squares and right-angled triangles
When squares are drawn using each side length of a right-angled triangle, something interesting happens.
For the triangle below:
(i) Use the side lengths in the triangle to create three squares.
(ii) Calculate the area of each square formed and write a relationship between them. Area
Area
Area
5 units4 units
3 units
5 units
4 units
3 units
3 units
4 units
5 units
2
1
3
1
2
3
4 4 4 16
3 3 3 9
5 5 5 25
2
2
2
#
#
#
= = =
= = =
= = =
Area of a square = (side length)2
Area 1 + Area 2 = Area 3
16 units2 + 9 units2 = 25 units2
7ISERIES TOPIC
5Pythagoras’ TheoremMathletics Passport © 3P Learning
How does it work? Your Turn Pythagoras’ Theorem
Squares and right-angled triangles
Show that the relationship between the areas of the squares formed using each side length works for these right-angled triangles:
1
2
3
Area 1 =
Area 2 =
Area 3 =
5 units13 units
12 units
10 units
Try the jigsaw puzzle at the back of this booklet to see another way of showing this property.
6 units
8 units
Area 1 =
Area 2 =
Area 3 =
1
3
2
13 units
12 units
5 units
3
2
110 units
6 units
8 units
SQUARES AND RIGHT- ANGLED TRIANGLES SQUAR
ES AND
RIGHT
- ANGLED TRIANGLES
..../...../20...
6 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
How does it work? Pythagoras’ Theorem
Pythagoras’ Theorem for right-angled triangles
The squares and right-angled triangles section showed that a relationship exists between the side lengths of right-angled triangles. This relationship is called Pythagoras’ Theorem.
Use Pythagoras’ Theorem to determine which of the following triangles are right-angled or not.
(i)
cother short sidehypotenuse
short side a
b
If the rule does not work, then it is not a right-angled triangle.
(short side)2 + (other short side)2 = (longest side)2 a2 + b2 = c2
always the hypotenuse
4 6 822 2+ =
16 36 64+ =
52 64!
0.8 1.5 1.72 2 2+ =
0.64 2.25 2.89+ =
2.89 2.89=
` not a right-angled triangle ` is a right-angled triangle
8
4
6
1.5
1.7
0.8
(ii)
Substitute lengths into Pythagoras’ Theorem
(short side)2 + (other short side)2 = (longest side)2
7ISERIES TOPIC
7Pythagoras’ TheoremMathletics Passport © 3P Learning
How does it work? Your Turn Pythagoras’ Theorem
Pythagoras’ Theorem for right-angled triangles
Use Pythagoras’ Theorem to calculate which of the following triangles are right-angled or not.1
15
12
9
a b
c d
Not right-angled
e f
Right-angled
25
2014
Not right-angledRight-angled
3.4
9.6
7.1
Not right-angledRight-angled
1.2
3.5
3.7
Not right-angledRight-angled
21
29
20
Not right-angledRight-angled
25
7
24
Not right-angledRight-angled
PYTHAGORAS’ THEOREM
FOR RIGHT-ANGLED TRIANGLES FOR RIGHT-ANGLED TRIANGLES
PYTHAGORAS’ THEOR
EM
ca
b
22
2=
+
..../...../20...
8 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
How does it work? Your Turn Pythagoras’ Theorem
Pythagoras’ Theorem for right-angled triangles
Name all the right-angled triangles pictured below and mark where the right-angle is with the correct symbol.
16
20
24 20
2
3
A
B
J
12
K I
J
10 10.5
14.5H
15AC N
M
L
21
29
48 48
20
52
H
G
K
Earn an awesome passport with this one! Name all the right-angled triangles in this image and markwhere the right-angles are with the correct symbol.
The right-angled triangles are:
The right-angled triangles are:
Remember, triangles are named by their vertices.
D
EF
= ΔDEF
Diagram not drawn to scale.
65R S
522436
155
1612
P QU
T
153280
V
..../...../20...
* AWESOME *
*
AWESOM
E *
7ISERIES TOPIC
9Pythagoras’ TheoremMathletics Passport © 3P Learning
How does it work? Your Turn Pythagoras’ Theorem
Pythagoras’ Theorem by measurement
For each of these right-angled triangles:
(i) Use a ruler to carefully measure the length of each side to the nearest whole millimetre. (ii) Use the measurements to complete the table at the bottom of the page.
ca
b
c a
b
c
a
b
c
a
b
c
a bc
a
b
12
3 4
6
5
PYTHAGORAS’THEOREM BY MEASUREMENT PYTH
AGOR
AS’T
HEOR
EM
B
Y MEASUREMENT
..../...../20...
1
2
3
4
5
6
a2a b2b c2c a2 + b2
10 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
Pythagoras’ TheoremWhere does it work?
Calculating the length of the hypotenuse
We can use Pythagoras’ Theorem to calculate the length of the hypotenuse if the two shorter sides of the right-angled triangle are already known.
The order that we put the short side values into the formula does not matter.
Calculate the length of the hypotenuse for this right-angled triangle.
Hypotenuse2 = short side2 + other short side2
Let’s label the hypotenuse ‘c’
24
7c
c
c
c
7 24
49 576
625
2 2 2
2
2
= +
= +
=
c
c
625
25
=
=
or c 24 72 2 2= + — short side order does not matter
To calculate the length c, square root this value
Write the positive answer only because it’s a length
Calculate the length of the hypotenuse for this right-angled triangle accurate to 2 decimal places.
Hypotenuse2 = short side2 + other short side2
8.16 units
m
. .
. .
.
m
m
m
8 16 3 14
66 5856 9 8596
76 4452
2 2 2
2
2
= +
= +
=
.
. ...
.
m
m
m
76 4452
8 743294574
8 74.
=
=
Label the hypotenuse for easy referencing
To calculate the length m, square root this value
Answer in square root form
Write full calculator reading before rounding
Approximate answer rounded to 2 decimal places
Rounded off decimal values are approximate answers only, so the '≈' symbol should be used.
3.14 units
`
Stop here if asked for answer in exact form
c a b2 2 2= +
units
units
`
`
`
`
` units
exact form
7ISERIES TOPIC
11Pythagoras’ TheoremMathletics Passport © 3P Learning
Where does it work? Your Turn Pythagoras’ Theorem
Calculating the length of the hypotenuse
Complete these Pythagoras’ Theorem calculations to find the length of the hypotenuse in each triangle.
1
a
6
8
c
c
c
c
62 2
2
2
2= +
= +
=
c
c
=
=
8
15
g
b
g
g
g
82 2 2
2
2
= +
= +
=
g
g
=
=
exact form
Use Pythagoras’ Theorem to calculate the length of the hypotenuse in each of these triangles.2
a b
c
d
`
`
`
`
`
`
`
`
5
12
1.6
1.2
exact form
units units
12 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Your Turn Pythagoras’ Theorem
Calculating the length of the hypotenuse
3 Calculate the length of the hypotenuse in each of these triangles, leaving answers in exact form.
a b
4 Calculate the length of the hypotenuse in each of these triangles, rounding answers to 2 decimal places. psst! Remember to use the ‘≈’ for rounded answers.
a b
1.1
6.0
h12
35
n
10 units 9 units
c
3.4 units
5.9 units
p
7ISERIES TOPIC
13Pythagoras’ TheoremMathletics Passport © 3P Learning
Where does it work? Your Turn Pythagoras’ Theorem
Calculating the length of the hypotenuse
Calculate the total length of the 3-stage flight path over the hills shown below accurate to 1 decimal place. psst! You need to do 3 hypotenuse calculations first.5
CALCUL
ATING THE LENGTH OF THE HYPOTENUSE
..../...../2
0...c
ab
2
2
2
=+
198m
39m
36mStage 3
Flight path
Stage 1
Launch pad
252m40m 360m
Stage 2
14 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Pythagoras’ Theorem
Calculating the length of a short side
To calculate a short side length in a right-angled triangle, the formula needs a little adjusting.
Subtract the given short side squared away from the hypotenuse squared.
Calculate the length of the missing side for this right-angled triangle.
Short side2 = hyponenuse2 - other short side2
Let’s label the short side ‘a’
15 unitsa
a
a
a
a
a
15 12
225 144
81
81
9
2 2 2
2
2
= -
= -
=
=
=
`
or a c b2 2 2= -
Always (longest side)2 – (smaller side)2
To calculate the length of a, square root this value
Write the positive answer only because it’s a length
Calculate the length of side k for this right-angled triangle, leaving answer in exact form.
Hypotenuse2 = short side2 + other short side2
7.3
k
. .
. .
.
.
k
k
k
k
7 3 1 9
53 29 3 61
49 68
49 68
2 2 2
2
2
= -
= -
=
=
`
or k c a2 2 2= -
Label the hypotenuse for referencing
To calculate the length k, square root this value
Answer in exact form
Answers left in square root form are not approximations, so the ‘=’ can still be used.
12 units
1.9
a c b2 2 2= -
units
units
units`
`
`
7ISERIES TOPIC
15Pythagoras’ TheoremMathletics Passport © 3P Learning
Where does it work? Your Turn Pythagoras’ Theorem
Calculating the length of a short side
Fill the gaps in these calculations to find the length of the missing short side in each triangle.1
a b
26
24
a
8.5
b1.3
5670
j
18.1 unitsa
18 unitsa b
a
a
a
262 2 2
2
2
= -
= -
=
.b
b
b
1 32 2 2
2
2
= -
= -
=
Use Pythagoras’ Theorem to calculate the length of the missing short side in each of these triangles.2
CALCUL
ATING THE LENGTH OF THE SHORT SI
DE ..../...../20...
ac
b2
2
2
=-
a
a
=
=
`
`
`
`
`
`
`
`
b
b
=
=units units
16 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Your Turn Pythagoras’ Theorem
Calculating the length of a short side
3 Calculate the length of the missing short side in each of these triangles, leaving answers in square root form.
11
17
b w
a b
a b
4 Calculate the length of the missing short side in each of these triangles, rounding answers to 1 decimal place.
psst! Remember to use the ‘. ’ for rounded off answers.
11.75
14.25
13.8
8.3
y
41.08
23.42
x
7ISERIES TOPIC
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Where does it work? Your Turn Pythagoras’ Theorem
Combination of hypotenuse and short side calculations
Match the triangles with the correct side length on the right to reveal the missing answer.
The special name given a right-angled triangle which is exactly one half of an equilateral triangle:
=triangle
15
20
545
544
42
42.1
41.9
53.2
32
68
20
14.16
30
67
g
h
e
d
b
a
c
MI
E
H
Q
E
14.12
2.9
73.4
25
67.7
33
60
COMBO T
IME
CO
MBO TIME COMBO TIME ..../...../
20...
1 2 3 4 5 6
2
1
3
4
5
6
18 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Pythagoras’ Theorem
Applications of Pythagoras’ Theorem
Distances that are difficult to measure can be solved using Pythagoras’ Theorem.
Calculate how far a 15 m support will reach up a wall if standing 9 m away from its base.
Let’s label the height up the wall ‘h’
15 m h
15 9
225 81
144
h
h
h
2 2 2
2
2
= -
= -
=
m12h =
Walls and buildings are perpendicular to the ground
This is a short side of a right-angled triangle
Square root this value to find h
Write the positive answer only because it’s a length
Calculate the perimeter of the garden shaped like a right-angled triangle shown below.
35 12
1225 144
1369
c
c
c
2 2 2
2
2
= +
= +
=
m
m
c
c
1369
37
=
=
First need the distance along the hypotenuse
Square root this value
Add all the side distances together
Pythagoras’ Theorem is often used to calculate unknown lengths in perimeter and area calculations.
9 m
Remember: Perimeter is the total distance around the outside
`
`
`
`
`
m m m
m
35 12 37
84
= + +
=
` Perimeter of the garden
Support
35 m
c12 m
7ISERIES TOPIC
19Pythagoras’ TheoremMathletics Passport © 3P Learning
Where does it work? Your Turn Pythagoras’ Theorem
Applications of Pythagoras’ Theorem
1 One end of a 13 m straight wire is attached to a flag pole 12 m above the ground. How far away from the base of the flag pole (x) will the other end be attached to the ground as a support?
3 To avoid going through a muddy swamp, Mila walks 1.7 km west and then 3.9 km South.
(i) How far is Mila away from where she started at the end of this walk? Round answer to 2 decimal places.
psst! West and South directions are perpendicular (90o) to each other.
APPLICATIONS
OF PY
THAGOR
AS’ THE
OREM
..../...
../20...
PYTHAGORAS’ THEOREM
APPLICATIONS OF
(ii) How much further did Mila have to walk to avoid the swamp?
Start1.7 km
Finish
3.9 km
Gini has made a pudding in a large 42 cm by 34 cm tray. If she first cuts the pudding diagonally from one corner to the other, how long was the cut Gini made to the nearest whole cm?
2
42 cm
34 cm
x
12 m13 m
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7ISERIES TOPIC
Where does it work? Your Turn Pythagoras’ Theorem
3.3 m
Applications of Pythagoras’ Theorem
4 (i) Calculate the base length of the painted triangle below. (ii) Use the base length to calculate the area of the triangle. psst! The area equals (base # height) ' 2
5 The mouse wants to run the shortest path from point A to point C across the floor shown. Calculate the shortest path between these two points if corner B blocks the direct path.
2.6 m17 m
6 (i) Calculate the length of the side marked ‘y’. (ii) Calculate the perimeter of the trapezium.
(i)
(ii)
C
18 m
54.4 m
A
B
172 cm
120 cm
137 cm
y
(i)
(ii)
Diagram not drawn to scale.
Base length
7ISERIES TOPIC
21Pythagoras’ TheoremMathletics Passport © 3P Learning
Where does it work? Your Turn Pythagoras’ Theorem
Applications of Pythagoras’ Theorem
Give these two trickier applications a go to earn an awesome passport stamp!
7 Use Pythagoras’ Theorem twice to find the distance between points X and Y. psst! Find the difference between WY and WX
8 Calculate the length of the cable support BD on the crane picture below if CD = 9.5 m, AB = 6 m and BC = 18.5 m.
*
AWESO
ME *
..../.....
/20...
* AWESOME *
65
34
16
Y
X
W
A
CB
6 m
18.5 m
9.5 m
D
Z
22 Pythagoras’ TheoremMathletics Passport © 3P Learning
7ISERIES TOPIC
Pythagoras’ TheoremWhat else can you do?
Pythagorean triads
Pythagorean triad is the special name given to a set of three positive integers that work in Pythagoras’ Theorem.
Pythagorean triad integers represent the side lengths of a right-angled triangle.
The integers 3, 4 and 5 are the best known Pythagorean triad.
5
Braces are used to display a set of integers
Integers work in Pythagoras’ Theorem
They form a right-angled triangle
Show whether these sets of integers form a Pythagorean triad or not.
?
?
24 8 22
576 64 484
576 548
2 2 2
!
= +
= +
You can use the LHS = RHS test approach here too
Pythagoras’ Theorem is used to show if a set of three integers form a Pythagorean triad.
Test: does
4
Remember: integers are just whole numbers.
` Is , ,8 22 24" , a Pythagorean triad?
, ,3 4 5" , is a Pythagorean triad because 3 4 52 2 2+ =
, ,3 4 5" ,
3
Because each integer is a side length for a right-angled triangle, negative values are not allowed.
Test to see if: (largest value)2 = (smallest value)2 + (middle value)2
Largest value
Middle value
Smallest value
(i) , ,8 22 24" , (ii) , ,9 12 15" ,
Largest value
Middle value
Smallest value
Test: does ?
?
15 9 12
225 81 144
225 225
2 2 2= +
= +
=
` Is , ,9 12 15" , a Pythagorean triad?
Yes No Yes No
15
12
9
=
=
=
24
22
8
=
=
=
7ISERIES TOPIC
23Pythagoras’ TheoremMathletics Passport © 3P Learning
What else can you do? Your Turn Pythagoras’ Theorem
Pythagorean triads
1 Write the side lengths of these right-angled triangles as a Pythagorean triad set.
2 Show whether these sets of positive integers form a Pythagorean triad or not.
PYTHAGOREA
N TRIADS PYTHAGOREAN TRIADS ..../...
../20...3
5
4
a , ,7 24 25" , b , ,14 48 50" , c , ,12 34 36" ,
d , ,15 36 39" , e , ,16 60 63" , f , ,12 30 31" ,
Yes No Yes No Yes No
Yes No Yes No Yes No
1220
16
35
1237
26
1024
941
40
psst! Note that they are written in order of size.
b
c d
12 ,16 ,20" , { , , }
{ , , }
{ , , }
a
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7ISERIES TOPIC
What else can you do? Pythagoras’ Theorem
Euclid’s formula for Pythagorean triads
Euclid of Alexandria (a Greek mathematician) developed this method to find most Pythagorean triads:
Step 1: Choose two positive integers p and q. When you pick these integers, make p larger than q i.e. p > q
Step 2: Substitute values for p and q into these to make a Pythagorean triad:
small integer other small integer largest integer
, 2 ,p q pq p q2 2 2 2- +" ,
Use the values p = 3 and q = 2 to make a Pythagorean triad.
Substitute in p and q values
Calculate final values
Integers form a Pythagorean triad
Use Euclid’s formula to make a Pythagorean triad that contains the number 8.
Here is another example with a specific request.
For the values p = 3 and q = 2
Let’s check that it works
`
, 2 ,p q pq p q2 2 2 2- +" ,
, ,9 4 12 9 4- +" ,
, ,5 12 13" ,
5 12 13 ?2 2 2+ =
25 144 169?+ =
169 169=
3 2 , 2 3 2 , 3 22 2 2 2# #- +" ,
This will be the easiest to use this time
p > q and ensures a value of 8
Substitute in p and q values
Calculate final values
Put values into ascending order for triad
For the values p = 3 and q = 2
, 2 ,p q pq p q2 2 2 2- +" ,
4 1 , 2 4 1, 4 12 2 2 2# #- +" ,
, ,15 8 17" ,
, ,8 15 17" ,
pq2 8=Let
` p = 4 and q = 1
, ,16 1 8 16 1- +" ,
7ISERIES TOPIC
25Pythagoras’ TheoremMathletics Passport © 3P Learning
What else can you do? Your Turn Pythagoras’ Theorem
Euclid’s formula for Pythagorean triads
1 Complete this table for the given values of p and q to make Pythagorean triads.
2 Make Pythagorean triads matching each of these specific requests.
(i) Find a Pythagorean triad in which p = 7 and p q2 2- is equal to 33.
EU
CLID’S FORMULA FOR PYTHAGOREAN
TRAIDS
..../...../20...
,,
pq
pq pq
2
22
22
-
+
"
,
*
(ii) Find a Pythagorean triad in which q = 5 and p q2 2+ is equal to 61.
p2 - q2 p 2pq p2 + q2 Triadq
2 1
3 1
5 2
7 6
11 3
21 18
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7ISERIES TOPIC
What else can you do? Your Turn Pythagoras’ Theorem
Remember me?
Euclid’s formula for Pythagorean triads
Find a group of three integers that includes the number 14 and forms a Pythagorean triad.
Use the space below to show why the value of p must be greater than the value of q when using Euclid’s formula to find a Pythagorean triad.
Use your own values of p and q to help show your answer.
hint: Pythagorean triads can be made using positive integers only.
This is definitely worth an awesome stamp!!
3
4
* AWESOME *
..../...../20...
* AWESOME
*
7ISERIES TOPIC
27Pythagoras’ TheoremMathletics Passport © 3P Learning
What else can you do? Pythagoras’ Theorem
Wheel of Theodorus
When squares are drawn using each side length of a right-angled triangle, something interesting happens.
How does this work?
Using Pythagoras’ Theorem:
Starting with this isosceles triangle, each new right-angled triangle is built using the hypotenuse of the previous one.
The length of the longest sides form a nice square root number pattern.
c a b
c a b
2 2 2
2 2
= +
= +`
` for the first triangle:
` for the second triangle:
c 1 1
2
2 2= +
=
( )c 1 2
1 2
3
2 2= +
= +
=
c
2
1 1
3
2
12
1
1 c
c
b
a
2
3
4 5
6
7
8
9
10
11
12
1
1
1 1
1
1
1
1
1
1
1
The pattern continues in this fashion always using 1 as the shortest side value.
1
1
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7ISERIES TOPIC
What else can you do? Your Turn Pythagoras’ Theorem
Wheel of Theodorus
Using the start made for you, continue the pattern always using 2 as the shortest side value to create your own neat spiral wheel.
8
WHEEL OF
THEODORUS WHEEL OF THEODORUS ..../
...../20..
.
2
2
2
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What else can you do? Your Turn Pythagoras’ Theorem
Reflection Time
Reflecting on the work covered within this booklet:
1 What useful skills have you gained by learning Pythagoras’ Theorem?
2 Write about one way you think you could apply Pythagoras’ Theorem to a real life situation.
3 If you discovered or learnt about any shortcuts to help with Pythagoras calculations or some other cool facts, jot them down here:
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Cheat Sheet Pythagoras’ Theorem
Here is a summary of the things you need to remember for Pythagoras’ Theorem
Right-angled triangles
• These special triangles all have a right-angle (angle of size 90o) as one of the internal angles.
• The 90o angle is always opposite the longest side.
• The two shorter sides are always perpendicular to each other.
Pythagoras’ Theorem for right-angled triangles
• The squares and right-angled triangles section showed that a relationship exists between the side lengths of right-angled triangles. This relationship is called Pythagoras’ Theorem.
• If the rule does not work, then it is not a right-angled triangle.
Calculating the length of the hypotenuse
• Use Pythagoras’ Theorem to calculate the length of the hypotenuse if the two shorter sides of the right-angled triangle are already known.
• The order that we put the short side values into the formula does not matter.
Calculating the length of a short side
• To calculate a short side length in a right-angled triangle, the formula needs a little adjusting.
• Subtract the given short side squared away from the hypotenuse squared.
Pythagorean Triads
• A Pythagorean triad is a set of three positive numbers that work in Pythagoras’ Theorem.
Making Pythagorean Triads
Step 1: Choose two positive numbers p and q.
When you pick these numbers, make p larger than q i.e. p > q
Step 2: Substitute values for p and q into this to make a Pythagorean triad: , 2 ,p q pq p q2 2 2 2- +" ,
Step 3: Write the values in ascending order.
cother short sidehypotenuse
short side a
b
(short side)2 + (other short side)2 = (longest side)2 a2 + b2 = c2
always the hypotenuse
a c b2 2 2= -
c a b2 2 2= +
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Cheat Sheet Pythagoras’ Theorem
Squares and right-angled triangles: Jigsaw Puzzle
Step 1: Cut the two shaded squares out from the page
Step 2: Cut the larger of these two along the dotted lines.
Step 3: Arrange all the pieces to fit perfectly inside this square.
Step 4: Stick the pieces to the page to show the area of the two smaller squares add together to give the area of this square on the hypotenuse.
Jigsaw Puzzle Pythagoras’ Theorem
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Pythagoras’ Theorem Notes
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