pyth-thm-workbook

(1)

Pythagoras’ Theorem

Mathlecs

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(2)

Pythagoras’ theorem

Student Book - Series I

Topics

Topic 1 -

Hypotenuse of the right angled triang

le

_{//__}

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 Pythagorea

n triads

_{//__}

Topic 5 -

Finding the length of the hypotenuse

_{//__}

Topic 6 -

Finding the length of one of the other sides

_{//__}

Topic 7 -

Miscellaneou

s questions on Pythagoras’ theorem

_{//__}

Topic 8 -

Problem solving and Pythagoras’ theorem

_{//__}

Author of The Topics and Topic Tests: AS Kalra

Practice Tests

Topic 1 - Topic test A

_{//__}

Topic 2 - Topic test B

(3)

Pythagoras’ theorem

T

opic 1:

Hypotenuse of the right

angled triangle

Q

QUESTIONUESTION 11

Name the hypotenuse of each right angled triangle.

a

a b b cc

d

d e e f f

Q

QUESTIONUESTION 22

Name the hypotenuse of the triangle named below in the diagram.

a

a b b cc

d

d e e f f

Q

QUESTIONUESTION 33

Complete the following statements.

a

a _{is the length of the side opposite to angle D.}_{is the length of the side opposite to angle D.}

b

b _{is the length of the side opposite to angle E.}_{is the length of the side opposite to angle E.}

c

c _{is the length of the}_{is the length of the}_{side opposite to angle F.}_{side opposite to angle F.}

d

d _{is the length}_{is the length}_{of the hypotenuse of}_{of the hypotenuse of}_ΔDEF_ΔDEF_._.

e

e _{is the area of the square on the side opposite to}_{is the area of the square on the side opposite to}_∠_∠_D._D.

f

f _{is the area of the square on the side opposite to}_{is the area of the square on the side opposite to}_∠_∠_F_F_._.

a a b b c c B B A A C C K K LL J J r r p p q q P P Q Q R R _l_l n n m m L L M M N N T T S S R R W W V V U U B B A A C C D D Q Q P P R R S S M M L L N N K K S S P P R R Q Q T T D D B B C C A A DD E E HH G G F F ∆ ∆ ADC ADC _∆_∆PSRPSR _∆_∆LJMLJM ∆ ∆PSTPST _∆_∆CBDCBD ∆ ∆FHGFHG E E FF D D d d e e f f E E T T JJ

(4)

Pythagoras’ theorem

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 1 2 2 33 4 4 5 5 66 7 7 8 8 99 a b a b cc aa22 _b_b22 _c_c22 _a_a22₊₊ _b_b22 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 4 4 3 3 B B A A C C 5 5 B B A A CC 12 12 13 13 5 5 _B_B A A CC 10 10 8 8 6 6 20 20 16 16 B B A A C C 12 12 B B A A C C 17 17 15 15 8 8 B B A A C C 12 12 15 15 9 9 26 26 24 24 B B A A C C 10 10 B B A A C C 20 20 25 25 ₁₅₁₅ _B_B A A C C 16 16 30 30 34 34

T

(5)

Pythagoras’ theorem

In the following right angled triangles, circle the correct statement.

1 1 aa _a_a22₌₌ _b_b22₊₊_c_c22 ₇₇ _a_a _s_s22₌₌_t_t22₊₊_u_u22 b b _b_b22₌₌ _a_a22₊₊_c_c22 _b_b _t_t22₌₌_s_s22₊₊_u_u22 c c _c_c22₌₌_a_a22₊₊_b_b22 _c_c _u_u22₌₌_s_s22₊₊_t_t22 2 2 aa _d_d22₌₌ _e_e22₊₊_f_f22 ₈₈ _a_a _x_x22₌₌_y_y22₊₊_z_z22 b b _e_e22₌₌_d_d22₊₊_f_f22 _b_b _y_y22₌₌_x_x22₊₊_z_z22 c c _f_f22₌₌ _d_d22₊₊_e_e22 _c_c _z_z22₌₌_x_x22₊₊_y_y22 3 3 aa _g_g22₌₌ _h_h22₊₊_i_i22 ₉₉ _a_a _u_u22₌₌_v_v22₊₊_w_w22 b b _h_h22₌₌ _g_g22₊₊_i_i22 _b_b _v_v22₌₌ _u_u22₊₊_w_w22 c c _i_i22₌₌_g_g22₊₊ _h_h22 _c_c _w_w22₌₌_u_u22₊₊_v_v22 4 4 aa _j_j22₌₌ _k_k22₊₊_l_l22 ₁₀₁₀ _a_a _b_b22₌₌_c_c22₊₊_d_d22 b b _k_k22₌₌_j_j22₊₊_l_l22 _b_b _c_c22₌₌ _b_b22₊₊_d_d22 c c _l_l22₌₌_j_j22₊₊_k_k22 _c_c _d_d22₌₌_b_b22₊₊_c_c22 5 5 aa _m_m22₌₌_n_n22₊₊_o_o22 ₁₁₁₁ _a_a _e_e22₌₌_f_f22₊₊_g_g22 b b _n_n22₌₌ _m_m22₊₊_o_o22 _b_b _f_f22₌₌_e_e22₊₊_g_g22 c c _o_o22₌₌ _m_m22₊₊_n_n22 _c_c _g_g22₌₌_e_e22₊₊_f_f22 6 6 aa _p_p22₌₌ _q_q22₊₊_r_r22 ₁₂₁₂ _a_a _h_h22₌₌_i_i22₊₊_j_j22 b b _q_q22₌₌_p_p22₊₊_r_r22 _b_b _i_i22₌₌_h_h22₊₊_j_j22 c c _r_r22₌₌_p_p22₊₊_q_q22 _c_c _j_j22₌₌_i_i22₊₊_h_h22 A A B B C C U U E E D D F F Z Z G G HH I I J J K K L L M M O O N N P P Q Q R R V V WW B B C C D D F F GG H H I I J J T T S S Y Y X X U U E E

Topic 3: Selecting the correct Pythagoras’ rule

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Pythagoras’ theorem

Q

QUESTIONUESTION

1

1 Use your calculator to ﬁnd the following squares.

Use your calculator to ﬁnd the following squares.

a

151522₌₌

_b

₁₃₁₃22₌₌

_c

₄₀₄₀22₌₌

d

282822₌₌

_e

₅₅22₌₌

_f

₆₉₆₉22₌₌

g

101022₌₌

_h

₁₇₁₇22₌₌

_i

₈₁₈₁22₌₌

j

8822₌₌

_k

₄₁₄₁22₌₌

_l

₉₉₉₉22₌₌ Q

QUESTIONUESTION

2

2 Use the square root key to ﬁnd

Use the square root key to ﬁnd

nn

.

a

nn22 = 169 = 169

b

nn22 = 841 = 841

c

nn22 = 576 = 576

d

nn22 = 4761 = 4761

e

nn22 = 100 = 100

f

nn22 = 1444 = 1444

g

nn22 = 144 = 144

h

nn22 = 441 = 441

i

nn22 = 1600 = 1600

j

nn22 = 2809 = 2809

k

nn22 = 784 = 784

l

nn22 = 5625 = 5625 Q

QUESTIONUESTION

3

3 Which of the following are Pythagorean triads?

Which of the following are Pythagorean triads?

a

{2, 4, 6}{2, 4, 6}

b

{9, 12, 15}{9, 12, 15}

c

{9, 40, 41}{9, 40, 41}

d

{4, 6, 10}{4, 6, 10}

e

{3, 4, 5}{3, 4, 5}

f

{8, 13, 17}{8, 13, 17}

g

{8, 10, 12}{8, 10, 12}

h

{19, 40, 41}{19, 40, 41}

i

{6, 8, 10}{6, 8, 10}

j

{5, 12, 13}{5, 12, 13}

k

{15, 36, 39}{15, 36, 39}

l

{8, 15, 17}{8, 15, 17} Q

QUESTIONUESTION

4

4 Prove that the following triangles are right angled triangles.

Prove that the following triangles are right angled triangles.

a a b b 3 3 4 4 5 5 13 13 5 5 12 12

Topic 4: Squares, square roots and Pythagorean triads

(7)

Pythagoras’ theorem

Q

QUESTIONUESTION

1

1 Find the length of the hypotenuse in each of the following. (All measurements are in

Find the length of the hypotenuse in each of the following. (All measurements are in

centimetres.)

a a bb c c dd e e f f Q

QUESTIONUESTION

2

2 Find the length of the hypotenuse correct to one decimal place. (All measurements are

Find the length of the hypotenuse correct to one decimal place. (All measurements are

in centimetres.)

a a bb c c dd e e f f 3 3 4 4 x x 6 6 88 x x 15 15 ₂₀₂₀ x x 10 10 24 24 x x 3 30 0 1166 x x x x 5 5 12 12 3 3 5 5 x x 4 4 7 7 x x 3.5 3.5 5 5 x x 4.2 4.2 7 7 x x 1.2 1.2 3.4 3.4 x x 3.5 3.5 6.5 6.5 x x

T

(8)

Pythagoras’ theorem

Q

QUESTIONUESTION

1

1 In the following triang

In the following triang

les, f

ind the length o

f the unknown sides. (All mea

surements are

in centimetres.)

QUESTIONUESTION

2

2 Find the length of the unknown side correct to one decimal place. (All measurements are

Find the length of the unknown side correct to one decimal place. (All measurements are

in centimetres.)

a a bb c c dd e e f f 12 12 13 13 x x 15 15 17 17 x x 41 41 40 40 x x 12 12 20 20 x x 6 6 10 10 x x 25 25 7 7 x x 15.3 15.3 8.9 8.9 x x 14 14 6 6 x x 17 17 9 9 x x 14.2 14.2 5 5 x x 4.5 4.5 12.3 12.3 x x 7 7 15 15 x x

T

(9)

Pythagoras’ theorem

Q

QUESTIONUESTION

1

1 In each of the following, fnd the length of the unknown sides (All measurements are in

In each of the following, fnd the length of the unknown sides (All measurements are in

centimetres.)

QUESTIONUESTION

2

2 Find the length of the unknown side correct to two decimal places. (All measurements

Find the length of the unknown side correct to two decimal places. (All measurements

are in centimetres.)

a a bb c c dd e e f f 5 5 12 12 x x 20 20 16 16 _x_x 12 12 20 20 x x 6 6 ₁₀₁₀ x x 15 15 21 21 x x 5 5 24 24 x x 7 7 xx 4 4 5 5 6 6 14 14 x x 3 3 12 12 x x 30 30 24 24 x x 2 2 3 3 x x y y 4 4 17 17 x x 8 8 11 11

Topic 7: Miscellaneous questions on Pythagoras’ theorem

(10)

Pythagoras’ theorem

1

1 Find the length Find the length of the of the diagonal of a diagonal of a square of side square of side length 5 cm. ________length 5 cm. ________________________________________________________________________

2

2 Find the length Find the length of the diagonal of the diagonal of a rectangle of a rectangle of length 35 cm and width 12 cmof length 35 cm and width 12 cm..

_________________________________________________

______________________________________________________________________________________________ _____________________________________________

3

3 What is the altitude What is the altitude of an equilateral triangle of an equilateral triangle whose sides are each 16 whose sides are each 16 cm long?cm long?

Give your answer correct to two decimal

Give your answer correct to two decimal places.places.

__________________________________________

4

4 A 15 metre ladder rests against a wall and A 15 metre ladder rests against a wall and its foot is 4 metres its foot is 4 metres away from the base of away from the base of the wall. How highthe wall. How high

does it reach up the wall? Give your answer correct to two decimal places.

_________________________________________________

______________________________________________________________________________________________ _____________________________________________

5

5 The sides of The sides of a rectangle are a rectangle are 12 12 cm and 6 cm. Find the length cm and 6 cm. Find the length of the diagonal.of the diagonal.

Give your answer correct to one decimal place.

_________________________________________________

______________________________________________________________________________________________ _____________________________________________

6

6 The hypotenuse of a right angled The hypotenuse of a right angled triangle is 30 triangle is 30 cm. If one of cm. If one of the shorter sides is 18the shorter sides is 18cm, nd the length of thecm, nd the length of the

other side.

_________________________________________________

______________________________________________________________________________________________ _____________________________________________

7

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.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

8 Find the length of the unknown side in each of the following triangles, correct to two decimal places. (AllFind the length of the unknown side in each of the following triangles, correct to two decimal places. (All

measurements are in centimetres.)

measurements are in centimetres.) a

a b b cc

d

d e e f f

9

9 Find the perimeter of the triangle below (correct to one decimal place) by nding the hypotenuse rst.Find the perimeter of the triangle below (correct to one decimal place) by nding the hypotenuse rst.

11 11 15 15 x x 30 30 29 29 12 12 18 18 a a 14 14 4 4 y y 12 12 9 9 x x 25 25 60 60 x x 36 36 64 64 x x

Topic 8: Problem solving and Pythagoras’ theorem

(11)

1

1 _{A triangle is said to satisfy the rule}_{A triangle is said to satisfy the rule} cc22₌₌ _a_a22₊₊_b_b22_{for which special triangle?}_{for which special triangle?}

A

acute angledacute angled

B

right angledright angled

C

obtuse angledobtuse angled

D

none of thesenone of these 2

2 _{The longest side of a right angled triangle is called the}_{The longest side of a right angled triangle is called the}

A

shortest sideshortest side

B

middle sidemiddle side

C

hypotenusehypotenuse

D

3 _{Given that}_{Given that}cc22₌₌ _a_a22₊₊_b_b22_and_and_a_a_{= 8,}_{= 8,} _b_b_{= 15, what is the value of}_{= 15, what is the value of} _c_c_?_?

A

1717

B

2323

C

289289

D

529529 4

4 _{Pythagoras’}_{Pythagoras’}_{theorem can be}_{theorem can be}_{applied to}_{applied to}

A

acute angled trianglesacute angled triangles

B

obtuse angled trianglesobtuse angled triangles

C

right angled trianglesright angled triangles

D

any triangleany triangle 5

5 _{The hypotenuse of a right angled}_{The hypotenuse of a right angled}_{triangle is 17}_{triangle is 17}_{cm. If one side}_{cm. If one side}_{is 15 cm, the third}_{is 15 cm, the third}_{side is}_{side is}

A

14 cm14 cm

B

12 cm12 cm

C

10 cm10 cm

D

8 cm8 cm 6

6 _{If two sides of a}_{If two sides of a}_{right angled triangle are 2.4}_{right angled triangle are 2.4}_{m and 1}_{m and 1}_{m then the hypotenuse is}_{m then the hypotenuse is}

A

2.4 m2.4 m

B

2.6 m2.6 m

C

3.4 m3.4 m

D

3.8 m3.8 m 7

7 _{The Pythagorean result for a triangle ABC right angled at}_{The Pythagorean result for a triangle ABC right angled at}_{C is}_{C is}

A

aa22₌₌_b_b22₊₊_c_c22

B

_b_b22₌₌_a_a22₊₊_c_c22

C

_c_c22₌₌_a_a22₊₊_b_b22

D

_{none of these}_{none of these}

8

8 _{The hypotenuse of a right angled triangle is opposite to the}_{The hypotenuse of a right angled triangle is opposite to the}

A

acute angleacute angle

B

right angleright angle

C

obtuse angleobtuse angle

D

9 _{If two shorter sides of a}_{If two shorter sides of a}_{right angled triangle are 7 m and 8}_{right angled triangle are 7 m and 8}_{m, then the hypotenuse is}_{m, then the hypotenuse is}

A

6565

B

8585

C

113113

D

193193 10

10 _{In a triangle ABC right angled at C, the}_{In a triangle ABC right angled at C, the}_{hypotenuse is named as}_{hypotenuse is named as}

A

aa

B

bb

C

cc

D

_{none of these}_{none of these}

11

11 _{If two sides of a right angled triangle are 6 cm and 8 cm, then the hypotenuse is}_{If two sides of a right angled triangle are 6 cm and 8 cm, then the hypotenuse is}

A

10 cm10 cm

B

9.4 cm9.4 cm

C

12 cm12 cm

D

14 cm14 cm 12

12 _If_Ifnn22_{= 2304 then}_{= 2304 then} _n_n_equals_equals

A

3838

B

4242

C

4848

D

5252 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Pythagoras’ theorem

Topic

Topic Test

Test

PART

PART A

A

Instructions

Instructions This part consists of 12 multiple-choice questionsThis part consists of 12 multiple-choice questions

Each question is worth 1 mark

Fill in only ONE

Fill in only ONE CIRCLE for each questionCIRCLE for each question

Calculators are NOT allowed

Time

Time allowed: allowed: 15 15 minutes minutes TTotal otal marks marks = = 1212

Marks

(12)

Marks

Pythagoras’ theorem

Topic

Topic Test

Test

PART

PART B

B

Instructions

Instructions This part consists of 15 questionsThis part consists of 15 questions

Each question is worth 1 mark

Write answers in the answers-only column

Time

Time allowed: allowed: 20 20 minutes minutes TTotal otal marks marks = = 1515

Questions

Questions Answers Answers onlyonly

1

1 _If_Ifnn22 = 3844 then nd the value of = 3844 then nd the value ofnn

2

2 _{Is {6, 8, 10} a Pythagorean triad?}_{Is {6, 8, 10} a Pythagorean triad?}

3

3 _{Prove that ΔPQR is a right}_{Prove that ΔPQR is a right}_angled_angled

triangle.

Find the length of the unknown side in the following triangles correct

to two decimal places.

4 4 5 5 66 7 7 8 8 99 10 10 11 11 1212 13 13 14 14 1515 P P Q Q R R 36 36 45 45 27 27 3 m 3 m 22 22 mm x x 9.6 cm 9.6 cm 7.2 cm 7.2 cm x x 17 m 17 m 15 m 15 m x x 2 m 2 m 1 1 mm x x 12 12 mm 13 13 mm x x 24 cm24 cm 7 7 cmcm x x 12 12 mm 5 5 mm x x 8 8 mm 11 11 mm x x 13 13 mm 3 3 mm x x 8 8 cmcm 15 cm 15 cm x x 23 cm 23 cm 11 cm 11 cm x x 20 20 mm 17 17 mm x x 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 1 1 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

pyth-thm-workbook

Pythagoras’ Theorem

Pythagoras’ Theorem

Mathlecs

Mathlecs

Instant

Instant

Workbooks

Workbooks

r

r

y

y

x

x

S

Pythagoras’ theorem

Pythagoras’ theorem

Student Book - Series I

Student Book - Series I

Contents

Contents

Topics

Topics

Topic 1 -

Topic 1 -

Hypotenuse of the right angled triang

Hypotenuse of the right angled triang

le

le

__/__/__

__/__/__

Date completed

Date completed

Topic 2 -

Topic 2 -

Naming the sides of a right angled triangle

Naming the sides of a right angled triangle

__/__/__

__/__/__

Topic 3 -

Topic 3 -

Selecting the correct Pythagoras’ rule

Selecting the correct Pythagoras’ rule

__/__/__

__/__/__

Topic 4 -

Topic 4 -

Squares, square roots and Pythagorea

Squares, square roots and Pythagorea

n triads

n triads

__/__/__

__/__/__

Topic 5 -

Topic 5 -

Finding the length of the hypotenuse

Finding the length of the hypotenuse

__/__/__

__/__/__

Topic 6 -

Topic 6 -

Finding the length of one of the other sides

Finding the length of one of the other sides

__/__/__

__/__/__

Topic 7 -

Topic 7 -

Miscellaneou

Miscellaneou

s questions on Pythagoras’ theorem

s questions on Pythagoras’ theorem

__/__/__

__/__/__

Topic 8 -

Topic 8 -

Problem solving and Pythagoras’ theorem

Problem solving and Pythagoras’ theorem

__/__/__

__/__/__

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_{//__}

_b

_b

_c

_c

_e

_e

_f

_f

_h

_h

_i

_i

_k

_k

_l

_l