GCSE Maths · Higher · Chapter 11: Right-angled triangles

Pythagoras' theorem and trigonometry in right-angled triangles

Grades 4–9About 85 minutesCalculator allowed93 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Work out $6. 5^{2}$ .
[1 mark]
Answer

A2
$x^{2} = 196$ and $x$ is positive. Find $x$ .
[1 mark]
Answer
A3
Solve $\frac{x}{7} = 0.6$ .
[1 mark]
Answer
A4
Solve $\frac{12}{x} = 0.8$ .
[2 marks]
Answer
A5
Round $23.4567$ to 3 significant figures.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

In triangle $A B C$ , angle $B = 9 0^{\circ}$ , $A B = 8$ cm and $B C = 13$ cm. Work out the length of $A C$ . Give your answer to 1 decimal place.

Hint

A C

is the hypotenuse: it is opposite the right angle. Square the two shorter sides and add.

B1
Work out the length $x$ . Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

Example 2

In triangle $P QR$ , angle $Q = 9 0^{\circ}$ , $P R = 17.2$ cm and $P Q = 9.8$ cm. Work out the length of $QR$ . Give your answer to 1 decimal place.

Hint This time the hypotenuse is given. Square it and take away the square of the other side.

B2
Work out the length $x$ . Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

Example 3

In triangle $A B C$ , angle $B = 9 0^{\circ}$ , angle $A = 3 8^{\circ}$ and $A C = 12$ cm. Work out the length of $B C$ . Give your answer to 3 significant figures.

Hint Label the sides from the angle you know:

B C

is opposite,

A C

is the hypotenuse. Opposite and hypotenuse means sine.

B3
Work out the length $x$ . Give your answer to 3 significant figures.
[3 marks]
Not drawn accurately
Answer

Example 4

In triangle $D E F$ , angle $E = 9 0^{\circ}$ , $D F = 11$ cm and $E F = 7$ cm. Work out the size of angle $D$ . Give your answer to 1 decimal place.

Hint

E F

is opposite angle

D

and

D F

is the hypotenuse. Write

sin D = \frac{7}{11}

, then use the inverse sine.

B4
Work out the size of angle $θ$ . Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
The diagram shows an isosceles triangle. The dashed line is its height $h$ .
Not drawn accurately
(a)
Work out $h$ .
[2 marks]
Answer
(b)
Work out the area of the triangle.
[2 marks]
Answer
Total for C1: 4 marks

C2
A triangle has sides of length 9 cm, 12 cm and 16 cm.
Show that the triangle is not right-angled.
[2 marks]
C3
$A$ is the point $(- 2, 7)$ and $B$ is the point $(6, - 8)$ .
Work out the length of the line segment $A B$ .
[3 marks]
Answer
C4
The diagram shows a cuboid 9 cm long, 4 cm wide and 7 cm high.
Work out the length of the diagonal $P Q$ . Give your answer to 3 significant figures.
[3 marks]
Not drawn accurately
Answer
C5
A ladder 6.5 m long leans against a vertical wall. The foot of the ladder is 1.8 m from the wall on level ground.
Work out the angle $θ$ between the ladder and the ground. Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer
C6
Leah stands 40 m from the foot of a vertical tower on level ground. The angle of elevation of the top of the tower from where she stands is $2 8^{\circ}$ .
Work out the height $h$ of the tower. Give your answer to 3 significant figures.
[3 marks]
Not drawn accurately
Answer
C7
A boat sails 12 km due north from a harbour $H$ , and then 7 km due east to a buoy $B$ .
Work out the bearing of $B$ from $H$ . Give your answer to the nearest degree.
[3 marks]
Answer
C8
Work out the length $x$ . Give your answer to 3 significant figures.
[3 marks]
Not drawn accurately
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
$A B C D$ is a rectangle. $A D = 7.2$ cm and the diagonal $A C = 19.5$ cm.
Work out the perimeter of the rectangle. Give your answer to 3 significant figures.
[3 marks]
Not drawn accurately
Answer

D2
Ravi works out the length $x$ . He writes " $x = 8^{2} + 1 5^{2} = 17$ cm".
Explain the mistake Ravi has made, and work out the correct value of $x$ . Give your answer to 3 significant figures.
[3 marks]
Not drawn accurately
Answer
D3
$T$ is the top of a vertical cliff 54 m high. Two boats, $A$ and $B$ , are in a straight line out to sea from the foot of the cliff.
The angle of depression of $A$ from $T$ is $3 2^{\circ}$ . The angle of depression of $B$ from $T$ is $1 9^{\circ}$ .
Work out the distance $A B$ . Give your answer to 3 significant figures.
[4 marks]
Not drawn accurately
Answer
D4
The diagram shows a cuboid 12 cm long, 5 cm wide and 7 cm high.
Work out the size of angle $G A F$ , the angle between the diagonal $A G$ and the base of the cuboid. Give your answer to 1 decimal place.
[4 marks]
Not drawn accurately
Answer
D5
A ship sails 18 km from a port $P$ to a point $Q$ on a bearing of $11 5^{\circ}$ .
Not drawn accurately
(a)
Work out how far east of $P$ the point $Q$ is. Give your answer to 3 significant figures.
[2 marks]
Answer
(b)
Work out how far south of $P$ the point $Q$ is. Give your answer to 3 significant figures.
[2 marks]
Answer
(c)
Write down the bearing of $P$ from $Q$ .
[1 mark]
Answer
Total for D5: 5 marks
D6
The diagram shows a quadrilateral $A B C D$ . Angle $A B C = 9 0^{\circ}$ and angle $A C D = 9 0^{\circ}$ .
$A B = 8.6$ cm, $C D = 5.3$ cm and angle $A C B = 4 1^{\circ}$ .
Not drawn accurately
(a)
Work out the length of $A D$ . Give your answer to 3 significant figures.
[4 marks]
Answer
(b)
Work out the size of angle $C A D$ . Give your answer to 1 decimal place.
[2 marks]
Answer
Total for D6: 6 marks
D7
A triangle has sides of length 2.8 cm, 4.5 cm and 5.3 cm.
Show that the triangle is right-angled.
[2 marks]
D8
A ladder 7.2 m long leans against a vertical wall on level ground. At first the ladder makes an angle of $6 8^{\circ}$ with the ground. The foot of the ladder is then moved closer to the wall, so that the angle becomes $7 5^{\circ}$ .
How much higher up the wall does the top of the ladder reach? Give your answer in centimetres to the nearest centimetre.
[4 marks]
Answer
D9
A pyramid has a square base of side 10 cm. Its top $V$ is vertically above the centre of the base, and its height is 12 cm.
Work out the length of the sloping edge $V A$ . Give your answer to 3 significant figures.
[4 marks]
Not drawn accurately
Answer
D10
An isosceles triangle has two sides of length 14 cm. The angle between them is $4 8^{\circ}$ .
Work out the length $x$ of the third side. Give your answer to 3 significant figures.
[3 marks]
Not drawn accurately
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
The diagonal from one corner of a cube to the opposite corner is 15 cm long.
Work out the total surface area of the cube.
[3 marks]
Hint 1 · What to try
Call the edge of the cube $s$ . Use Pythagoras twice to write the long diagonal in terms of $s$ .
Hint 2 · The first line
$s^{2} + s^{2} + s^{2} = 1 5^{2}$
Hint 3 · The full method
$3 s^{2} = 225$ , so $s^{2} = 75$ . Each of the six faces has area $s^{2}$ , so the surface area is $6 \times 75 = 450$ cm².
Answer

E2
In triangle $A B C$ , angle $B = 9 0^{\circ}$ and $A C = 30$ cm. The tangent of angle $A$ is $0.75$ .
Work out the perimeter of the triangle.
[3 marks]
Hint 1 · What to try
$tan A = 0.75 = \frac{3}{4}$ . Draw a right-angled triangle with sides 3 and 4 next to angle $A$ .
Hint 2 · The first line
That triangle has hypotenuse 5, so triangle $A B C$ is 6 times bigger.
Hint 3 · The full method
$B C = 18$ cm and $A B = 24$ cm, so the perimeter is $18 + 24 + 30 = 72$ cm.
Answer
E3
Two points $P$ and $Q$ are 30 m apart on level ground, in a straight line with the foot of a vertical mast. The angle of elevation of the top of the mast is $2 6^{\circ}$ from $P$ and $4 1^{\circ}$ from $Q$ .
Work out the height of the mast. Give your answer to 3 significant figures.
[4 marks]
Hint 1 · What to try
Call the height $h$ . Write the distance from each point to the foot of the mast in terms of $h$ .
Hint 2 · The first line
$\frac{h}{tan 2 6 ^{\circ}} - \frac{h}{tan 4 1 ^{\circ}} = 30$
Hint 3 · The full method
$h (\frac{1}{tan 2 6 ^{\circ}} - \frac{1}{tan 4 1 ^{\circ}}) = 30$ , so $h = \frac{30}{2.0503 - 1.1504} = 33.3$ m.
Answer
E4
A box is a cuboid with a base 60 cm by 45 cm. A thin rod 110 cm long must fit inside the box.
Work out the smallest whole number of centimetres the height of the box can be.
[3 marks]
Hint 1 · What to try
The longest rod that fits runs from one corner to the opposite corner.
Hint 2 · The first line
$6 0^{2} + 4 5^{2} + h^{2} = 11 0^{2}$
Hint 3 · The full method
$h^{2} = 6475$ , so $h = 80.47 \dots$ The height must be at least this, so the smallest whole number of centimetres is 81.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use Pythagoras' theorem to find the hypotenuse or a shorter side, and to test whether a triangle is right-angled.
I can use Pythagoras' theorem twice to find a length inside a cuboid or a pyramid.
I can choose sine, cosine or tangent and use it to find a side or an angle.
I can solve problems with angles of elevation and depression, bearings and isosceles triangles.

Answers and mark scheme

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