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Pythagoras’ Theorem
Mathletics Instant
Workbooks
ry
x
Student Book - Series I-1
Pythagoras’ theorem
Student Book - Series I
Contents
Topics
Topic 1 - Hypotenuse of the right angled triangle __/__/__ Date completed
Topic 2 - Naming the sides of a right angled triangle __/__/__ Topic 3 - Selecting the correct Pythagoras’ rule __/__/__ Topic 4 - Squares, square roots and Pythagorean triads __/__/__ Topic 5 - Finding the length of the hypotenuse __/__/__ Topic 6 - Finding the length of one of the other sides __/__/__ Topic 7 - Miscellaneous questions on Pythagoras’ theorem __/__/__ Topic 8 - Problem solving and Pythagoras’ theorem __/__/__
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Author of The Topics and Topic Tests: AS Kalra
Practice Tests
Topic 1 - Topic test A __/__/__ Topic 2 - Topic test B __/__/__
1
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Topic 1: Hypotenuse of the right angled triangle
12 EXCEL ESSENTIAL SKILLS: YEAR 8 MATHEMATICS REVISION AND EXAM WORKBOOK 1
Pythagoras’ theorem EXCEL MATHEMATICS YEAR 8
pages 124–125UNIT 1: Hypotenuse of the right angled triangle
Question 1 Name the hypotenuse of each right angled triangle.a b c
d e f
Question 2 Name the hypotenuse of the triangle named below in the diagram.a b c
d e f
Question 3 Complete the following statements.
a is the length of the side opposite to angle D.
b is the length of the side opposite to angle E.
c is the length of the side opposite to angle F.
d is the length of the hypotenuse of ΔDEF.
e is the area of the square on the side opposite to ∠D.
f is the area of the square on the side opposite to ∠F.
Chapter 2
a
bc
B
A
C
K L
J
r
p
q
P
QR l
nm
L
M
N
T
SR
W
V
U
BA
CD
QP
RS
ML
NK
SP
RQ
T
DB
C
A D
E H
G
F
∆ADC ∆PSR ∆LJM
∆PST ∆CBD
∆FHG
E F
D
d
ef
E T J
2
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Pythagoras’ theorem EXCEL MATHEMATICS YEAR 8
pages 124–125
Chapter 2: Pythagoras’ theorem 13
For each of the following triangles, complete the table below and verify that the square on the hypotenuse is equal to the sum of the squares on the other two sides.
1 2 3
4 5 6
7 8 9
a b c a2 b2 c2 a2 + b2
1
2
3
4
5
6
7
8
9
UNIT 2: Naming the sides of a right angled triangle
4
3
B
AC
5
B
A C
1213
5 B
A C
10
8
6
20
16
B
A
C
12
B
A
C
17
15
8
BA
C
12
15
9
26
24
BA
C
10
B
A
C20
25 15 B
A
C
16
30
34
Topic 2: Naming the sides of a right angled triangle
3
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
14 EXCEL ESSENTIAL SKILLS: YEAR 8 MATHEMATICS REVISION AND EXAM WORKBOOK 1
Pythagoras’ theorem
In the following right angled triangles, circle the correct statement.1 a a2 = b2 + c2 7 a s2 = t2 + u2 b b2 = a2 + c2 b t2 = s2 + u2
c c2 = a2 + b2 c u2 = s2 + t2
2 a d2 = e2 + f 2 8 a x2 = y2 + z2 b e2 = d2 + f 2 b y2 = x2 + z2
c f 2 = d2 + e2 c z2 = x2 + y2
3 a g2 = h2 + i2 9 a u2 = v2 + w2 b h2 = g2 + i2 b v2 = u2 + w2
c i2 = g2 + h2 c w2 = u2 + v2
4 a j 2 = k2 + l 2 10 a b2 = c2 + d2 b k2 = j2 + l 2 b c2 = b2 + d2
c l2 = j2 + k2 c d2 = b2 + c2
5 a m2 = n2 + o2 11 a e2 = f 2 + g2 b n2 = m2 + o2 b f 2 = e2 + g2
c o2 = m2 + n2 c g2 = e2 + f 2
6 a p2 = q2 + r2 12 a h2 = i2 + j 2 b q2 = p2 + r2 b i2 = h2 + j 2 c r2 = p2 + q2 c j 2 = i2 + h2
UNIT 3: Selecting the correct Pythagoras’ rule
EXCEL MATHEMATICS YEAR 8
page 125
A
BC
U
E
D
F
ZG H
I
J
KL
M
ON
P
Q
R
V W
B
C
D
F G
H
I
J
TS
YX
U
E
Topic 3: Selecting the correct Pythagoras’ rule
4
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Pythagoras’ theorem
Chapter 2: Pythagoras’ theorem 15
Question 1 Use your calculator to find the following squares.
a 152 = b 132 = c 402 =
d 282 = e 52 = f 692 =
g 102 = h 172 = i 812 =
j 82 = k 412 = l 992 =
Question 2 Use the square root key to find n.
a n2 = 169 b n2 = 841 c n2 = 576
d n2 = 4761 e n2 = 100 f n2 = 1444
g n2 = 144 h n2 = 441 i n2 = 1600
j n2 = 2809 k n2 = 784 l n2 = 5625
Question 3 Which of the following are Pythagorean triads?
a {2, 4, 6} b {9, 12, 15} c {9, 40, 41}
d {4, 6, 10} e {3, 4, 5} f {8, 13, 17}
g {8, 10, 12} h {19, 40, 41} i {6, 8, 10}
j {5, 12, 13} k {15, 36, 39} l {8, 15, 17}
Question 4 Prove that the following triangles are right angled triangles.
a
b
UNIT 4: Squares, square roots and Pythagorean triads
3
45
13
512
EXCEL MATHEMATICS YEAR 8
pages 35–36, 125, 128–129
Topic 4: Squares, square roots and Pythagorean triads
5
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
16 EXCEL ESSENTIAL SKILLS: YEAR 8 MATHEMATICS REVISION AND EXAM WORKBOOK 1
Pythagoras’ theoremUNIT 5: Finding the length of the hypotenuse
Question 1 Find the length of the hypotenuse in each of the following. (All measurements are in centimetres.)
a b
c d
e f
Question 2 Find the length of the hypotenuse correct to one decimal place. (All measurements are in centimetres.)
a b
c d
e f
3
4x
6 8
x
15 20
x
10
24
x
30 16
x
x5
12
3
5
x
4
7
x
3.5
5
x
4.2
7
x
1.23.4
x
3.5
6.5
x
EXCEL MATHEMATICS YEAR 8
pages 125–126
Topic 5: Finding the length of the hypotenuse
6
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Pythagoras’ theorem
Chapter 2: Pythagoras’ theorem 17
Question 1 In the following triangles, f ind the length of the unknown sides. (All measurements are in centimetres.)
a b
c d
e f
Question 2 Find the length of the unknown side correct to one decimal place. (All measurements are in centimetres.)
a b
c d
e f
UNIT 6: Finding the length of one of the other sides
12
13
x
15
17
x
41
40
x
1220
x
6
10
x
25
7 x
15.3
8.9
x
14
6
x
17
9 x
14.2
5
x
4.5
12.3
x715
x
EXCEL MATHEMATICS YEAR 8
pages 126–127
Topic 6: Finding the length of one of the other sides
7
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
18 EXCEL ESSENTIAL SKILLS: YEAR 8 MATHEMATICS REVISION AND EXAM WORKBOOK 1
Pythagoras’ theorem
Question 1 In each of the following, fnd the length of the unknown sides (All measurements are in centimetres.)
a b
c d
e f
Question 2 Find the length of the unknown side correct to two decimal places. (All measurements are in centimetres.)
a b
c d
e f
UNIT 7: Miscellaneous questions on Pythagoras’ theorem
512
x
20
16 x
12
20
x
6 10
x
15
21
x
524
x
7 x
4
5
6
14
x
3
12
x
30
24x
2
3
x
y
4
17
x
8
11
EXCEL MATHEMATICS YEAR 8
pages 129–131
Topic 7: Miscellaneous questions on Pythagoras’ theorem
8
Pythagoras’ theorem
Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Pythagoras’ theorem
Chapter 2: Pythagoras’ theorem 19
1 Find the length of the diagonal of a square of side length 5 cm. ________________________________________
2 Find the length of the diagonal of a rectangle of length 35 cm and width 12 cm.
______________________________________________________________________________________________
3 What is the altitude of an equilateral triangle whose sides are each 16 cm long? Give your answer correct to two decimal places. __________________________________________
4 A 15 metre ladder rests against a wall and its foot is 4 metres away from the base of the wall. How high does it reach up the wall? Give your answer correct to two decimal places.
______________________________________________________________________________________________
5 The sides of a rectangle are 12 cm and 6 cm. Find the length of the diagonal. Give your answer correct to one decimal place.
______________________________________________________________________________________________
6 The hypotenuse of a right angled triangle is 30 cm. If one of the shorter sides is 18 cm, find the length of the other side.
______________________________________________________________________________________________
7 In a right angled triangle, the longest side is 39 cm and the shortest side is 15 cm. Find the length of the third side.
__________________________________________________________________________________________________
8 Find the length of the unknown side in each of the following triangles, correct to two decimal places. (All measurements are in centimetres.)
a b c
d e f
9 Find the perimeter of the triangle below (correct to one decimal place) by finding the hypotenuse first.
UNIT 8: Problem solving and Pythagoras’ theorem
11
15x
30
29
1218
a14
4
y
129
x
25
60
x
3664
x
EXCEL MATHEMATICS YEAR 8
pages 131–132
Topic 8: Problem solving and Pythagoras’ theorem
Total marks achieved for PART A 12
Pythagoras’ theoremTOPIC TEST PART AInstructions •This part consists of 12 multiple choice questions •Each question is worth 1 mark • Fill in only ONE CIRCLE for each question • Calculators may be used
Time allowed: 15 minutes Total marks = 12
Total marks achieved for PART A 12
Marks
20 EXCEL ESSENTIAL SKILLS: YEAR 8 MATHEMATICS REVISION AND EXAM WORKBOOK 1
1 A triangle is said to satisfy the rule c2 = a2 + b2 for which special triangle?
Aacute angled Bright angled C obtuse angled Dnone of these
2 The longest side of a right angled triangle is called the
Ashortest side Bmiddle side C hypotenuse Dnone of these
3 Given that c2 = a2 + b2 and a = 8, b = 15, what is the value of c?
A17 B23 C 289 D529
4 Pythagoras’ theorem can be applied to
Aacute angled triangles B obtuse angled triangles
Cright angled triangles D any triangle
5 The hypotenuse of a right angled triangle is 17 cm. If one side is 15 cm, the third side is
A14 cm B12 cm C 10 cm D8 cm
6 If two sides of a right angled triangle are 2.4 m and 1 m then the hypotenuse is
A2.4 m B2.6 m C 3.4 m D3.8 m
7 The Pythagorean result for a triangle ABC right angled at C is
Aa2 = b2 + c2 Bb2 = a2 + c2 C c2 = a2 + b2 Dnone of these
8 The hypotenuse of a right angled triangle is opposite to the
Aacute angle Bright angle C obtuse angle Dnone of these
9 If two shorter sides of a right angled triangle are 7 m and 8 m, then the hypotenuse is
A 65 B 85 C 113 D 193
10 In a triangle ABC right angled at C, the hypotenuse is named as
Aa Bb C c Dnone of these
11 If two sides of a right angled triangle are 6 cm and 8 cm, then the hypotenuse is
A10 cm B9.4 cm C 12 cm D14 cm
12 If n2 = 2304 then n equals
A38 B42 C 48 D52
1
1
1
1
1
1
1
1
1
1
1
1
Pythagoras’ theoremTOPIC TEST PART AInstructions •This part consists of 12 multiple choice questions •Each question is worth 1 mark • Fill in only ONE CIRCLE for each question • Calculators may be used
Time allowed: 15 minutes Total marks = 12
Total marks achieved for PART A 12
Marks
20 EXCEL ESSENTIAL SKILLS: YEAR 8 MATHEMATICS REVISION AND EXAM WORKBOOK 1
1 A triangle is said to satisfy the rule c2 = a2 + b2 for which special triangle?
Aacute angled Bright angled C obtuse angled Dnone of these
2 The longest side of a right angled triangle is called the
Ashortest side Bmiddle side C hypotenuse Dnone of these
3 Given that c2 = a2 + b2 and a = 8, b = 15, what is the value of c?
A17 B23 C 289 D529
4 Pythagoras’ theorem can be applied to
Aacute angled triangles B obtuse angled triangles
Cright angled triangles D any triangle
5 The hypotenuse of a right angled triangle is 17 cm. If one side is 15 cm, the third side is
A14 cm B12 cm C 10 cm D8 cm
6 If two sides of a right angled triangle are 2.4 m and 1 m then the hypotenuse is
A2.4 m B2.6 m C 3.4 m D3.8 m
7 The Pythagorean result for a triangle ABC right angled at C is
Aa2 = b2 + c2 Bb2 = a2 + c2 C c2 = a2 + b2 Dnone of these
8 The hypotenuse of a right angled triangle is opposite to the
Aacute angle Bright angle C obtuse angle Dnone of these
9 If two shorter sides of a right angled triangle are 7 m and 8 m, then the hypotenuse is
A 65 B 85 C 113 D 193
10 In a triangle ABC right angled at C, the hypotenuse is named as
Aa Bb C c Dnone of these
11 If two sides of a right angled triangle are 6 cm and 8 cm, then the hypotenuse is
A10 cm B9.4 cm C 12 cm D14 cm
12 If n2 = 2304 then n equals
A38 B42 C 48 D52
1
1
1
1
1
1
1
1
1
1
1
1
9Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Pythagoras’ theoremTopic Test PART AInstructions This part consists of 12 multiple-choice questions Each question is worth 1 mark Fill in only ONE CIRCLE for each question Calculators are NOT allowed
Time allowed: 15 minutes Total marks = 12
Marks
Total marks achieved for PART B 15
Marks
Pythagoras’ theoremTopic Test PART B
10Pythagoras’ theoremMathletics Instant Workbooks – Series I Copyright © 3P Learning
Instructions This part consists of 15 questions Each question is worth 1 mark Write answers in the answers-only columnTime allowed: 20 minutes Total marks = 15
Questions Answers only
Marks
Pythagoras’ theorem TOPIC TEST PART B
Total marks achieved for PART B 30
Chapter 2: Pythagoras’ theorem 21
Instructions •This part consists of 15 questions •Each question is worth 2 marks • Write answers in the answers-only column • Calculators may be used
Time allowed: 15 minutes Total marks = 30 Questions Answers only Marks
1 If n2 = 3844 then find the value of n
2 Is {6, 8, 10} a Pythagorean triad?
3 Prove that ΔPQR is a right angled triangle.
Find the length of the unknown side in the following triangles correct to two decimal places.
4 5 6
7 8 9
10 11 12
13 14 15
P
Q
R
36
45
27
3 m22 m
x9.6 cm
7.2 cm
x17 m
15 m
x
2 m
1 m
x
12 m
13 m
x 24 cm
7 cm
x
12 m
5 m
x8 m
11 m
x13 m
3 m
x
8 cm
15 cm
x23 cm
11 cmx
20 m
17 m
x
14
15
16
17
18
19
110
111
112
113
114
115
11
12
13
Marks
Pythagoras’ theorem TOPIC TEST PART B
Total marks achieved for PART B 30
Chapter 2: Pythagoras’ theorem 21
Instructions •This part consists of 15 questions •Each question is worth 2 marks • Write answers in the answers-only column • Calculators may be used
Time allowed: 15 minutes Total marks = 30 Questions Answers only Marks
1 If n2 = 3844 then find the value of n
2 Is {6, 8, 10} a Pythagorean triad?
3 Prove that ΔPQR is a right angled triangle.
Find the length of the unknown side in the following triangles correct to two decimal places.
4 5 6
7 8 9
10 11 12
13 14 15
P
Q
R
36
45
27
3 m22 m
x9.6 cm
7.2 cm
x17 m
15 m
x
2 m
1 m
x
12 m
13 m
x 24 cm
7 cm
x
12 m
5 m
x8 m
11 m
x13 m
3 m
x
8 cm
15 cm
x23 cm
11 cmx
20 m
17 m
x
14
15
16
17
18
19
110
111
112
113
114
115
11
12
13
Marks
Pythagoras’ theorem TOPIC TEST PART B
Total marks achieved for PART B 30
Chapter 2: Pythagoras’ theorem 21
Instructions •This part consists of 15 questions •Each question is worth 2 marks • Write answers in the answers-only column • Calculators may be used
Time allowed: 15 minutes Total marks = 30 Questions Answers only Marks
1 If n2 = 3844 then find the value of n
2 Is {6, 8, 10} a Pythagorean triad?
3 Prove that ΔPQR is a right angled triangle.
Find the length of the unknown side in the following triangles correct to two decimal places.
4 5 6
7 8 9
10 11 12
13 14 15
P
Q
R
36
45
27
3 m22 m
x9.6 cm
7.2 cm
x17 m
15 m
x
2 m
1 m
x
12 m
13 m
x 24 cm
7 cm
x
12 m
5 m
x8 m
11 m
x13 m
3 m
x
8 cm
15 cm
x23 cm
11 cmx
20 m
17 m
x
14
15
16
17
18
19
110
111
112
113
114
115
11
12
13