�3 16.1–16.2 Pythagoras’ Theorem

337A

1 Find the length of the hypotenuse in each triangle, correct to 3 signi� cant � gures.

a b

.................................................................. ..................................................................

c d

.................................................................. ..................................................................

2 An extending ladder is placed 9.5 m from the base of a building.

Calculate the length of the ladder when it reaches these heights on the building wall.

a 4.7 m .................................................................. b 7.4 m .................................................................

c 12.9 m .................................................................. d 17 m ..................................................................

Guided practice worksheet

(hypotenuse)2 = (short side)2 + (other short side)2

a2 = 142 + 92 = 196 + 81 = 277a = √277 = 16.64331698 (write more than 3 signi� cant � gures from your calculator…)a = 16.6 cm (3 s.f.) (… then round your answer to the required degree of accuracy)

14 cm

9 cm

a

7.5 cm

5.3 cm

25 mm

20 mm

7 cm

3 cm7.5 cm

5.3 cm

1 m

1 m

25 mm

20 mm

9.5 m

1 m

1 m

C

Questions are targeted at the grades indicated

�3 16.1–16.2 Pythagoras’ Theorem

337B

Guided practice worksheet

3 Find the length of the side marked with a letter, correct to 3 signi� cant � gures.

a b

.................................................................. ..................................................................

c d

.................................................................. ..................................................................

4 Calculate the height of each isosceles triangle, correct to the nearest millimetre (1 d.p.).

a b

.................................................................. ..................................................................

c d

.................................................................. ..................................................................

(short side)2 = (hypotenuse)2 – (other short side)2

a2 = 5.22 – 2.32 = 27.04 – 5.29 = 21.75a = √21.75 = 4.663689527a = 4.66 cm (3 sf) 5.2 cm

2.3 cm

a

73 mm

55 mm

a2.3 cm

4.6 cm

b

100 mm0.8 m

0.5 m300 mm

c

d

73 mm

55 mm

a2.3 cm

4.6 cm

b

100 mm0.8 m

0.5 m300 mm

c

d

73 mm

55 mm

a2.3 cm

4.6 cm

b

100 mm0.8 m

0.5 m300 mm

c

d

73 mm

55 mm

a2.3 cm

4.6 cm

b

100 mm0.8 m

0.5 m300 mm

c

d

Hint Divide each isosceles triangle into two identical right-angled triangles.

1.8 cm

1.8 cm

0.8 cm20 cm 20 cm

12 cm

2.5 cm

4.5 cm

2.5 cm 8 cm 8 cm

8 cm

1.8 cm

1.8 cm

0.8 cm20 cm 20 cm

12 cm

2.5 cm

4.5 cm

2.5 cm 8 cm 8 cm

8 cm

1.8 cm

1.8 cm

0.8 cm20 cm 20 cm

12 cm

2.5 cm

4.5 cm

2.5 cm 8 cm 8 cm

8 cm

1.8 cm

1.8 cm

0.8 cm20 cm 20 cm

12 cm

2.5 cm

4.5 cm

2.5 cm 8 cm 8 cm

8 cm

C

�3 16.1–16.2 Pythagoras’ Theorem

337C

Guided practice worksheet

5 The top of the leaning tower of Pisa is about 56.3 m high. The tower is about 56.4 m long. How far to the right has the tower leaned?

The tower has leaned ..................................................................

6 Karen can adjust the length of the guy rope to hold the tent pole upright.

a How far from the tent pole is the tent peg when the guy rope is i 3 m ii 2.6 m iii 2.2 m iv 2 m long?

i ..................................................................

ii ..................................................................

iii ..................................................................

iv ..................................................................

b How long is the guy rope when the tent peg is i 2.5 m ii 3 m iii 1.7 m from the tent pole?

i ..................................................................

ii ..................................................................

iii ..................................................................

56.4 m56.3 m

d

Guy rope Tent

1.2 m

?

Tent pole

Tent peg

C

�3 16.3 Finding the length of a line segment

339A

Guided practice worksheet

1 Find the length of each line segment, correct to 3 signi� cant � gures.

a b c

.................................................................. .................................................................. ..................................................................

d e f

.................................................................. .................................................................. ..................................................................

For the line segment AB:

Di� erence in x-coordinates = 5 – 1 = 4Di� erence in y-coordinates = 4 – 2 = 2AB2 = 42 + 22 = 16 + 4 = 20AB = √20 = 4.472135955 = 4.47 (3 s.f.)

y

0 x

4 – 2 = 2

5 – 1 = 4B(5,2)

A(1,4)

y

0 x

A

B

y

0 xD (4,1)

C (3,9)y

0

2

x

E

F (5,7)

y

0

10

xA

H (8,4)

Gy

0 xJ (5,7)

K (20,24)

(4,7)

(1,2)

y

0 5

7

x

M

L

y

0 x

A

B

y

0 xD (4,1)

C (3,9)y

0

2

x

E

F (5,7)

y

0

10

xA

H (8,4)

Gy

0 xJ (5,7)

K (20,24)

(4,7)

(1,2)

y

0 5

7

x

M

L

Questions are targeted at the grades indicated

C

�3 16.3 Finding the length of a line segment

339B

Guided practice worksheet

2 Find the length of each line segment, correct to 1 decimal place.

a b

............................................................................................ ............................................................................................

c d

............................................................................................ ............................................................................................

For the line segment AB:

Di� erence in x-coordinates = 3 – –4 = 3 + 4 = 7Di� erence in y-coordinates = 2 – –3 = 2 + 3 = 5AB2 = 52 + 72 = 25 + 49 = 74AB = √74 = 8.602325267 = 8.6 (1 d.p.)

y

0 x

A (–4,2)

2 – –3 = 5

B (3,–3)

3 – –4 = 7

y

0 x

A (–4,1)

B (3,5)

y

0 x

D (4,–2)

C (1,5)

y

0 x

E (–6,–6)

F (2,–2)

y

0 x

G (–5,1)

H (2,–5)

y

0 x

A (–4,1)

B (3,5)

y

0 x

D (4,–2)

C (1,5)

y

0 x

E (–6,–6)

F (2,–2)

y

0 x

G (–5,1)

H (2,–5)

C

�3 16.4–16.5 Using trigonometry in right-angled triangles

341A

Guided practice worksheet

1 Use your calculator to � nd the value of the following, correct to 4 decimal places, where necessary.

a sin 30° ............................................................................. b tan 10° .............................................................................

c cos 90° ............................................................................. d sin 28° .............................................................................

e cos 1° ............................................................................. f tan 88.2° .............................................................................

2 Use your calculator to � nd the value of x, correct to 1 decimal place, where necessary.

a tan x = 1 ............................................................................. b sin x = 0.7 .............................................................................

c cos x = 0.25 ............................................................................. d sin x = 0 .............................................................................

e tan x = 3.1 ............................................................................. f cos x = 0.05 .............................................................................

g sin x = 0.216 ............................................................................. h tan x = 33

.............................................................................

3 Calculate the size of each marked angle, correct to 1 decimal place.

a b

.................................................................. ..................................................................

c d

.................................................................. ..................................................................

Hint Use the [sin–1] key of your calculator

Hint Use SOHCAHTOA

The two sides we know are Adjacent (4.2 cm) and AdjHypotenuse (6.3 cm), so use cos x = ––– Hyp 4.2cos x = ––– = 0.666… 6.3x = 48.18968511 ... (press [sin–1] to � nd angle x)x = 48.2° (1 d.p.)

x

4.2 cm

6.3 cm

a

7 mm

9 mm

b2.5 cm

1 cm

c

100 m75 mm

d

0.6 cm 0.7 cm

a

7 mm

9 mm

b2.5 cm

1 cm

c

100 m75 mm

d

0.6 cm 0.7 cm

a

7 mm

9 mm

b2.5 cm

1 cm

c

100 m75 mm

d

0.6 cm 0.7 cm

a

7 mm

9 mm

b2.5 cm

1 cm

c

100 m75 mm

d

0.6 cm 0.7 cm

B

Questions are targeted at the grades indicated

C

�3 16.4–16.5 Using trigonometry in right-angled triangles

341B

Guided practice worksheet

4 Find the length of each side marked with a letter, correct to 3 signi� cant � gures.

a b

.................................................................. ..................................................................

c d

.................................................................. ..................................................................

x is the side Opposite the angle 40° and 15 mm is the OppAdjacent side, so use tan x = ––– Adj xtan 40° = ––– 15 x0.839099631... = ––– 1515 × 0.839099631 ... = x (multiply both sides by 15)x = 12.6 cm (3 s.f.)

15 mm

x

40°

18 cm

a

30°

4.2 m

b

50°

100 mm

c

d

26°

0.25 m

81°

18 cm

a

30°

4.2 m

b

50°

100 mm

c

d

26°

0.25 m

81°

18 cm

a

30°

4.2 m

b

50°

100 mm

c

d

26°

0.25 m

81°

18 cm

a

30°

4.2 m

b

50°

100 mm

c

d

26°

0.25 m

81°

B

�3 16.4–16.5 Using trigonometry in right-angled triangles

341C

Guided practice worksheet

5 Trevor drove from Cardiff to Leeds and then � ew due north from Leeds to Newcastle.Use the dotted right-angled triangles to answer the following questions.

a How far is Leeds

i north of Cardiff?

...........................................................................................................................

ii east of Cardiff?

...........................................................................................................................

b How far is Newcastle north of Cardiff?

...........................................................................................................................

c What is the bearing of Newcastle from Cardiff?

...........................................................................................................................

d What is the distance from Cardiff to Newcastle?

...........................................................................................................................

x

25°174 miles

Cardiff

Leeds

89 miles

Newcastle

Hint Find angle x

B

�3 16.4–16.5 Using trigonometry in right-angled triangles

341D

Guided practice worksheet

6 Ahmed downloaded these plans to make a kite.

a Use trigonometry to calculate

i length x

...........................................................................................................................

ii angle y

...........................................................................................................................

b Use Pythagoras’ Theorem to calculate the perimeter of the kite

...........................................................................................................................

Hint Calculate the hypotenuses and use the symmetry of a kite.

80 cm

40 cm

x

y

25°

B