Pyth
agor
as’ T
heor
em PYTHAGORAS’ THEOREMPYTHAGORAS’ THEOREMPYTHAGORAS’ THEOREM
www.mathletics.com.au
7ISERIES TOPIC
1Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
How does it work? Pythagoras’ TheoremSolutions
2
M
N
L
a b
Hypotenuse is side: y
Hypotenuse is side: DF
Hypotenuse is side: PQ
Hypotenuse is side: k
a b
c d
y
Q
RP
D
F
E
x
z
k
jl
Hypotenuse is side: a Hypotenuse is side: MN
Name the hypotenuse for each of these badly drawn triangles.
Page 3 questions
Right-angled triangles
ca
b
1
2 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
How does it work? Pythagoras’ TheoremSolutions
1
2
Area 1 = 5 units # 5 units = 25 units2
Area 2 = 12 units # 12 units = 144 units2
Area 3 = 13 units # 13 units = 169 units2
Area 1 + Area 2 = 25 units2 + 144 units2
Area 1 = 6 units # 6 units = 36 units2
Area 2 = 8 units # 8 units = 64 units2
Area 3 = 10 units # 10 units = 100 units2
Area 1 + Area 2 = 36 units2 + 64 units2
1
3
2
13 units
12 units
5 units
3
2
110 units
6 units
8 units
Page 5 questions
Squares and right-angled triangles
= 169 units2
= Area 3
= 100 units2
= Area 3
7ISERIES TOPIC
3Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
How does it work? Pythagoras’ TheoremSolutions
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
Page 7 questions
Pythagoras’ Theorem for right-angled triangles
1
9 12 81 144 15 225
225
2 2 2+ = + =
=
14 20 196 400 25 625
596
2 2 2+ = + =
=
7.1 3.4 50.41 11.56 9.6 92.16
.61 97
2 2 2+ = + =
=
1.2 3.5 1.44 12.25 3.7 13.69
.13 69
2 2 2+ = + =
=
20 21 400 441 29 841
841
2 2 2+ = + =
=
24 7 576 49 25 625
625
2 2 2+ = + =
=
4 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
How does it work? Pythagoras’ TheoremSolutions
16
20
25 20
A
B
J
12
KI
J
1010.5
14.5H
15AC
48
20
52
H
G
K
The right-angled triangles are: ΔAJK , ΔHIJ , ΔGHK
Page 8 questions
Pythagoras’ Theorem for right-angled triangles
2
12 16 20
400 400
2 2 2+ =
=
. .
. .
10 10 5 14 5
210 25 210 25
2 2 2+ =
=
20 15 24
625 576
2 2 2
!
+ =
N
M
L
21
29
48
29 21 48
1282 2304
2 2 2
!
+ =
48 20 52
2704 2704
2 2 2+ =
=
7ISERIES TOPIC
5Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
How does it work? Pythagoras’ TheoremSolutions
3 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.
65R S
522436
155
1612
P QU
T
153280
V
The right-angled triangles are:
ΔPUV
ΔQRU
ΔRSU
ΔSTU
15 36 39
1521 1521
2 2 2+ =
=
39 52 65
4225 4225
2 2 2+ =
=
24 52 3280
3280 3280
2 2 2+ =
=
12 16 20
400 400
2 2 2+ =
=
Page 9 questions
Pythagoras’ Theorem for right-angled triangles
Assuming the scale of the page is the same as the original print, the measurements should be as follows:NOTE: if not the same scale, the same relationship between your measurements should work.
Page 8 questions
Pythagoras’ Theorem for right-angled triangles
202545 mm 360060 mm 562575 mm 2025 + 3600 = 5625
490070 mm 57624 mm 547674 mm 4900 + 576 = 5476
22515 mm 129636 mm 152139 mm 225 + 1296 = 1521
1
2
3
4
5
6
129636 mm 592977 mm 722585 mm 1296 + 5929 = 7225
819 mm 160040 mm 168141 mm 81 + 1600 = 1681
160040 mm 176442 mm 336458 mm 1600 + 1764 = 3364
a2a b2b c2c a2 + b2
6 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Pythagoras’ TheoremSolutions
1 a 6 8
36 64
100
c
c
c
c
c
100
10
2 2 2
2
2
= +
= +
=
=
=
`
`
`
`
b 8 15
64 225
289
g
g
g
g
g
289
17
2 2 2
2
2
= +
= +
=
=
=
`
`
`
`
2 a b
Page 11 questions
Calculating the length of the hypotenuse
5 12
25 144
169
c
c
c
c
c
169
13
2 2 2
2
2
= +
= +
=
=
=
`
`
`
`
1.2 1.6
1.44 2.56
4
d
d
d
d
d
4
2
2 2 2
2
2
= +
= +
=
=
=
`
`
`
`
Page 12 questions
Calculating the length of the hypotenuse
a 1.1 6.0
1.21 36
37.21
.
h
h
h
h 37 21
2 2 2
2
2
= +
= +
=
=
`
`
`
b 12 35
144 1225
1389
n
n
n
n 1389
2 2 2
2
2
= +
= +
=
=
`
`
`
a bunits units
units units
units
units
units
( ) ( )
. ...
.
c
c
c
c
c
c
10 9
100 81
181
181
13 45362405
13 45
2 2 2
2 2 2
2 2
.
= +
= +
=
=
=
`
`
`
`
`
`
`
`
4
3
units units
units units
units
units
units
( . ) ( . )
. .
.
.
. ...
.
p
p
p
p
p
p
5 9 3 4
34 81 11 56
46 37
46 37
6 809552114
6 81
2 2 2
2 2 2 2
2 2
.
= +
= +
=
=
=
units to 2 decimal placesunits to 2 decimal places
in exact square root formin exact square root form
``
7ISERIES TOPIC
7Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
Where does it work? Pythagoras’ TheoremSolutions
a b
2
Page 13 questions
Calculating the length of the hypotenuse
5
m
40 198
1600 39204
40804
d
d
d
d
d
40804
202
2 2 2
2
2
= +
= +
=
=
=
Stage 1
m
39 252
1521 63504
65025
d
d
d
d
d
65025
255
2 2 2
2
2
= +
= +
=
=
=
Stage 2
m
m
. ...
.
d
d
d
d
d
d
36 360
1296 129600
130896
130896
361 7955224
361 8
2 2 2
2
2
.
= +
= +
=
=
=
Stage 3
`
`
`
`
`
`
`
`
`
`
`
`
` The total length of the 3 stage flight path 202 255 361.8. + + m
m818.8.
1
Page 15 questions
Calculating the length of a short side
26 24
676 576
100
a
a
a
a
a
100
10
2 2 2
2
2
= -
= -
=
=
=
8.5 1.3
72.25 1.69
70.56
.
.
b
b
b
b
b
70 56
8 4
2 2 2
2
2
= -
= -
=
=
=
`
`
`
`
`
`
`
`
a b70 56
4900 3136
1764
j
j
j
j
j
1764
42
2 2 2
2
2
= -
= -
=
=
=
units units
units units
units
units
units
(18.1 ) (18 )
327.61 324
3.61
.
.
b
b
b
b
b
3 61
1 9
2 2 2
2 2 2
2
= -
= -
=
=
=
`
`
`
`
`
`
`
to 2 decimal places
`
`
8 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Pythagoras’ TheoremSolutions
a b
. .
. .
.
.
.
.
x
x
x
x
x
x
41 08 23 42
1687 5664 548 4968
1139 07
1139 07
33 7501
33 8
2 2 2
2
2
.
= -
= -
=
=
=
`
`
`
`
3
Page 16 questions
Calculating the length of a short side
17 11
289 121
168
b
b
b
b 168
2 2 2
2
2
= -
= -
=
=
`
`
`
14.25 11.75
203.0625 138.0625
65
w
w
w
w 65
2 2 2
2
2
= +
= +
=
=
`
`
`
4 . .
. .
.
.
. ...
.
y
y
y
y
y
y
13 8 8 3
190 44 68 89
121 55
121 55
11 02497166
11 0
2 2 2
2
2
.
= -
= -
=
=
=
`
`
`
`
to 1 decimal point
a b
` to 1 decimal point
in exact square root formin exact square root form
`
7ISERIES TOPIC
9Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
Where does it work? Pythagoras’ TheoremSolutions
= 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
14.12
2.9
73.4
25
67.7
33
60
1 2 3 4 5 6
Page 17 questions
Combination of hypotenuse and short side calculations
The special name given a right-angled triangle which is exactly one half of an equilateral triangle:
H E M I E Q
MI
E
H
Q
E
2
1
3
4
5
6
10 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Pythagoras’ TheoremSolutions
3
(ii) To avoid the swamp, Mila walked 3.9 km + 1.7 km = 5.6 km
x
12 m13 m
Start1.7 km
Finish
3.9 km
2
Page 19 questions
Applications of Pythagoras’ Theorem
1
cm cm
cm cm
cm
cm
cm
(42 ) (34 )
1764 1156
2920
54.03702434
cut
cut
cut
cut
cut
cut
2920
54
2 2 2
2 2 2
2 2
.
= +
= +
=
=
=
`
`
`
`
`
m m
m m
m
m
m
(13 ) (12 )
169 144
25
5
x
x
x
x
x
25
2 2 2
2 2 2
2 2
= -
= -
=
=
=
`
`
`
`
to nearest whole cm
km km
km km
km
km
km
km
(1.7 ) (3.9 )
2.89 15.21
18.1
.
4.254409477 ...
4.25
d
d
d
d
d
d
18 1
2 2 2
2 2 2
2 2
.
= +
= +
=
=
=
`
`
`
`
` to 2 decimal points
(i)
` Mila walked a further 5.6 km - 4.25 km . 1.35 km
d
42 cm
34 cm cut
7ISERIES TOPIC
11Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
Where does it work? Pythagoras’ TheoremSolutions
3.3 m
2.6 m17 m
(i)
(ii)
C
18 m
54.4 m
A
B
Diagram not drawn to scale.
base
4
Page 20 questions
Applications of Pythagoras’ Theorem
m m
m m
m
m
m
(17 ) (2.6 )
289 6.76
282.24
.
16.8
base
base
base
base
base
282 24
2 2 2
2 2 2
2 2
= -
= -
=
=
=
`
`
`
`
m m
m
m
( . . )
43.68 2
.
16 8 2 6 2
21 84
2
2
# '
'
=
=
=
`
`
Area = (base # height) ' 2
Area
Area
Area`
5
AB m m
AB m m
AB m
AB m
AB m
(18 ) (3.3 )
324 10.89
334.89
.
18.3
334 89
2 2 2
2 2 2
2 2
= +
= +
=
=
=
`
`
`
`
BC m m
BC m m
BC m
BC m
BC m
(54.4 ) (3.3 )
2959.36 10.89
2970.25
.
54.5
2970 25
2 2 2
2 2 2
2 2
= +
= +
=
=
=
`
`
`
`
Distance AC AB BC
m m
m
18.3 54.5
72.8
= +
= +
=
Distance around wall = 79m
Shortest path
12 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
Where does it work? Pythagoras’ TheoremSolutions
6 172 cm
120 cm
137 cm
y
Page 20 questions
Applications of Pythagoras’ Theorem
cm cm cm
cm cm
cm cm
cm
cm
cm
(172 137 ) (120 )
(35 ) (120 )
1225 14400
15625
125
y
y
y
y
y
y
15625
2 2 2
2 2 2
2 2 2
2 2
= - +
= +
= +
=
=
=
`
`
`
`
(i)
(ii) Perimeter of the trapezium cm cm cm cm
cm
172 120 137 125
554
= + + +
=
WY YZ WZ
WY
WY 63
65 16
4225 256
3969
3969
2 2 2
2 2
= -
= -
= -
=
=
=65
34
16
Y
X
W
Page 21 questions
Applications of Pythagoras’ Theorem
7
WX XZ WZ
WX
WX 30
34 16
1156 256
900
900
2 2 2
2 2
= -
= -
= -
=
=
=
XY WY WX
63 30
33
= -
= -
=
` `
7ISERIES TOPIC
13Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
Where does it work? Pythagoras’ TheoremSolutions
A
C
B
6 m
18.5 m
9.5 m
D
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
AC BC AB
AC
AC
18.5 6
.
.
.
.
342 25 36
306 25
306 25
17 5
2 2 2
2 2
= -
= -
= -
=
=
=
AD AC DC
. .17 5 9 5
8
= -
= -
=
`
BD AD AB
BD
BD
8 6
64 36
100
100
10
2 2 2
2 2
= +
= +
= +
=
=
=
`
Page 21 questions
Applications of Pythagoras’ Theorem
m
m
m
14 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
What else can you do? Pythagoras’ TheoremSolutions
1
2 Show whether these sets of positive integers form a Pythagorean triad or not.
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" , , ,10 24 26" ,
, ,12 35 37" , , ,9 40 41" ,
?
?
7 24 25
49 576 625
625 625
2 2 2+ =
+ =
=
?
?
14 48 50
196 2304 2500
2500 2500
2 2 2+ =
+ =
=
?
?
12 34 36
144 1156 1296
1300 1296
2 2 2
!
+ =
+ =
?
?
15 36 39
225 1296 1521
1521 1521
2 2 2+ =
+ =
=
?
?
16 60 63
256 3600 3969
3856 3969
2 2 2
!
+ =
+ =
?
?
12 30 31
144 900 961
1044 961
2 2 2
!
+ =
+ =
Page 23 questions
Pythagorean triads
a
7ISERIES TOPIC
15Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
What else can you do? Pythagoras’ TheoremSolutions
2 (i) Find a Pythagorean triad in which p = 7 and p q2 2- is equal to 33
(ii) Find a Pythagorean triad in which q = 5 and p q2 2+ is equal to 61
2 1 32 2- = 2 2 1 4# # = 2 1 52 2
+ = { , , }3 4 5
3 1 82 2- = 2 13 6# # = 3 1 102 2
- = { , , }6 8 10
5 2 212 2- = 2 5 2 20# # = 5 2 292 2
+ = { , , }20 21 29
7 6 132 2- = 2 7 6 84# # = 7 6 852 2
+ = { , , }13 84 85
11 3 1122 2- = 2 11 3 66# # = 11 3 1302 2
+ = { , , }66 112 130
21 18 1172 2- = 2 2 11 8 756# # = 21 18 7562 2
+ = { , , }117 756 765
p2 - q2 p 2pq p2 + q2 Triadq
2 1
3 1
5 2
7 6
11 3
21 18
Page 25 questions
Euclid’s formula for Pythagorean triads
1
33
7 33
49 33
49 33
16
p q
q
q
q
q
q 4
2 2
2 2
2
2
2
- =
- =
- =
- =
=
=
2 2 7 4pq
56
# #=
=
7 4p q
65
2 2 2 2+ = +
=
Pythagorean triad is { 33 , 56 , 65 }
p q
p
p
p
p
p
61
5 61
25 61
61 25
36
6
2 2
2 2
2
2
2
- =
- =
- =
= -
=
=
pq2 2 6 5
60
# #=
=
6 5p q
11
2 2 2 2+ = -
=
Pythagorean triad is { 11 , 60 , 61 }
`
`
16 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
What else can you do? Pythagoras’ TheoremSolutions
Find a group of three integers that includes the number 14 and forms a Pythagorean triad.
hint: Pythagorean triads can be made using positive integers only.
Formal explanation:
3
4
Page 26 questions
Pythagorean triads
{ , 2 , }p q pq p q2 2 2 2- +
pq
p q
p q
2 14
2 14
7
# #
#
=
=
=
p 7= ( )q p q1 2=and
7 1p q
50
2 2 2 2+ = +
=
7 1p q
48
2 2 2 2- = -
=
` `
Pythagorean triad is: { , , }14 48 50
{ , 2 , }p q pq p q2 2 2 2- +
small integer other small integer largest integer
Showing using chosen values 2p = and 1q = :
From hint, Pythagorean triads are made using positive integers. ie. positive whole numbers only.One of the smaller integers is found using p q2 2
-
If the value of p was smaller than the value of q, then the answer would be negative. So this could not be used because only positive whole numbers are allowed.
When 2p = and 1q = , p q 2 1 32 2 2 2- = - = (this is a positive integer and is allowed)
If we swap these around, so 2p = and 2q = p q 1 2 32 2 2 2
- = - = - (this is a negative integer and is not allowed)
This will always happen if the value of p is smaller than the value of q when using Euclid’s formula.
Negative numbers are not allowed because each integer represents the length of the side of a right-angled triangle. So a side length of -3 does not make sense.
7ISERIES TOPIC
17Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
What else can you do? Pythagoras’ TheoremSolutions
Page 28 questions
Wheel of Theodorus
812
16
20
24
2
2
2 2
2
2
2
and so on
18 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
What else can you do? Pythagoras’ TheoremSolutionsJigsaw Puzzle Pythagoras’ TheoremSolutions
2
1
3
4
4
3
2
1
5
55
Page 31 questions
Squares and right-angled triangles: Jigsaw Puzzle
7ISERIES TOPIC
19Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
Pythagoras’ Theorem Notes
20 Pythagoras’ Theorem Solutions
Mathletics Passport © 3P Learning
7ISERIES TOPIC
Pythagoras’ Theorem Notes
APPLICATIONS
OF
PYTHA
GORAS’
THEOR
EM
..../...
../20...
PYTHAGORAS’ THEOREM
APPLICATIONS OF
RIGHT-ANGLED
TRIANG
LES RIGHT-ANGLED
TRIANGLES
..../...../20...
*
AWESO
ME *
..../.....
/20...
* AWESOME *
SQUARES AND RIGHT- ANGLED TRIANGLES SQUARES AND RIGHT- AN
GLED
TRI
ANGL
ES
..../...../20...
EU
CLID’S FORMULA FOR PYTHAGOREAN
TRAIDS
..../...../20...
, ,p q pq p q
22
22
2-
+
",
*