�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