Pythagoras’ Theorem - Holy Trinity Catholic School Division · 2 Pythagoras’ theorem ... UNIT 3: Selecting the correct Pythagoras’ rule EXCEL MATHEMATICS YEAR 8 ... Pythagoras’

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Pythagoras’ Theorem

Mathletics Instant

Workbooks

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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



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 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





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





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


pages 35–36, 125, 128–129

Topic 4: Squares, square roots and Pythagorean triads

5




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


pages 125–126

Topic 5: Finding the length of the hypotenuse

6





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


pages 126–127

Topic 6: Finding the length of one of the other sides

7





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


pages 129–131

Topic 7: Miscellaneous questions on Pythagoras’ theorem

8





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


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


Marks


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



Marks


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


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



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










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










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

Documents

Pythagoras’ Theorem - Holy Trinity Catholic School Division · 2 Pythagoras’ theorem ... UNIT 3: Selecting the correct Pythagoras’ rule EXCEL MATHEMATICS YEAR 8 ... Pythagoras’