GCSE Maths · Higher · Chapter 20: Properties of circles

Circle theorems, tangents, chords and the alternate segment

Grades 4–9About 85 minutesNo calculator90 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Two angles of a triangle are $4 3^{\circ}$ and $7 6^{\circ}$ . Work out the third angle.
[1 mark]
Answer

A2
An isosceles triangle has an angle of $4 0^{\circ}$ between its two equal sides. Work out the size of each of the other two angles.
[1 mark]
Answer
A3
Two angles on a straight line are $11 8^{\circ}$ and $x^{\circ}$ . Work out $x$ .
[1 mark]
Answer
A4
A right-angled triangle has shorter sides of 9 cm and 12 cm. Work out the length of the longest side.
[2 marks]
Answer
A5
Three angles of a quadrilateral are $9 5^{\circ}$ , $7 0^{\circ}$ and $11 0^{\circ}$ . Work out the fourth angle.
[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

$A$ , $B$ and $C$ are points on a circle with centre $O$ . $C$ is on the major arc $A B$ . Angle $A O B = 12 4^{\circ}$ . Work out angle $A C B$ . Give a reason.

Not drawn accurately

Hint The angle at the centre is twice the angle at the circumference when both stand on the same arc.

B1
$A$ , $B$ and $C$ are points on a circle with centre $O$ . $C$ is on the major arc $A B$ . Angle $A C B = 4 7^{\circ}$ . Work out angle $A O B$ . Give a reason.
[2 marks]
Answer

Example 2

$A B C D$ is a cyclic quadrilateral. Angle $D A B = 7 2^{\circ}$ . Work out angle $B C D$ . Give a reason.

Not drawn accurately

Hint

A

and

C

are opposite corners, and opposite angles of a cyclic quadrilateral add up to

18 0^{\circ}

B2
$P QR S$ is a cyclic quadrilateral. Angle $S P Q = 3 x$ and angle $QR S = 2 x + 3 0^{\circ}$ . Work out the value of $x$ . Give a reason.
[3 marks]
Answer $x = \dots\dots$

Example 3

$T P$ is a tangent to a circle with centre $O$ . $P$ is on the circle. $O P = 5$ cm and $O T = 13$ cm. Work out the length of $P T$ .

Not drawn accurately

Hint A tangent meets the radius at

9 0^{\circ}

, so triangle

O P T

is right-angled at

P

B3
$T P$ is a tangent to a circle with centre $O$ . $P$ is on the circle. $O P = 7$ cm and $P T = 24$ cm. Work out the length of $O T$ .
[2 marks]
Answer

Example 4

$T A$ is a tangent to a circle at $A$ . $A B$ and $A C$ are chords, and $C$ is on the other side of $A B$ from $T$ . Angle $T A B = 5 8^{\circ}$ . Work out angle $A C B$ . Give a reason.

Not drawn accurately

Hint The angle between a tangent and a chord equals the angle in the alternate segment.

B4
$T A$ is a tangent to the circle at $A$ . Angle $T A B = 6 4^{\circ}$ and angle $B A C = 4 5^{\circ}$ . Work out angle $A B C$ . Give reasons.
[3 marks]
Not drawn accurately
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
$A B$ is a diameter of the circle with centre $O$ . $C$ is a point on the circle and angle $C A B = 3 4^{\circ}$ . Work out angle $A B C$ . Give reasons.
[3 marks]
Not drawn accurately
Answer

C2
$A$ , $B$ , $C$ and $D$ are points on the circle. Angle $A D B = 4 1^{\circ}$ . Work out angle $A C B$ . Give a reason.
[2 marks]
Not drawn accurately
Answer
C3
$A$ , $B$ and $C$ are points on the circle with centre $O$ . Angle $O A B = 2 3^{\circ}$ .
Not drawn accurately
(a)
Work out angle $A O B$ .
[2 marks]
Answer
(b)
Work out angle $A C B$ . Give a reason.
[2 marks]
Answer
Total for C3: 4 marks
C4
$A B C D$ is a cyclic quadrilateral. $A B$ is extended to $E$ . Angle $C B E = 7 9^{\circ}$ . Work out angle $A D C$ . Give reasons.
[3 marks]
Not drawn accurately
Answer
C5
$P A$ and $P B$ are tangents to the circle with centre $O$ . Angle $A P B = 4 8^{\circ}$ .
Not drawn accurately
(a)
Work out angle $A O B$ . Give a reason.
[2 marks]
Answer
(b)
Work out angle $P A B$ . Give a reason.
[2 marks]
Answer
Total for C5: 4 marks
C6
A chord of a circle is 16 cm long. The radius of the circle is 10 cm. Work out the distance from the centre of the circle to the chord.
[2 marks]
Answer
C7
$T A$ is a tangent to the circle with centre $O$ . $A B$ is a chord and angle $T A B = 6 8^{\circ}$ . Work out angle $A O B$ . Give reasons.
[3 marks]
Not drawn accurately
Answer
C8
$P$ is a point 20 cm from the centre of a circle of radius 12 cm. A tangent is drawn from $P$ to the circle. Work out the length of the tangent.
[2 marks]
Answer
C9
$A$ , $B$ , $C$ and $D$ are points on the circle. The chords $A C$ and $B D$ meet at $X$ . Angle $A B D = 3 8^{\circ}$ and angle $B D C = 4 9^{\circ}$ . Work out angle $C X D$ . Give a reason.
[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
$A$ , $B$ , $C$ and $D$ are points on the circle with centre $O$ . Angle $A B C = 11 6^{\circ}$ . Work out angle $A O C$ . You must give a reason for each stage of your working.
[3 marks]
Not drawn accurately
Answer

D2
Lily says, "Angle $A D C = 6 8^{\circ}$ , because angles in the same segment are equal."
$A B C D$ is a cyclic quadrilateral and angle $A B C = 6 8^{\circ}$ . Lily is wrong. Explain why, and work out angle $A D C$ .
[2 marks]
Not drawn accurately
Answer
D3
$T A$ is a tangent to the circle with centre $O$ . $A$ , $B$ and $C$ are points on the circle. Angle $T A C = 3 x$ and angle $A B C = 2 x + 1 8^{\circ}$ .
Not drawn accurately
(a)
Work out the value of $x$ . Give a reason.
[3 marks]
Answer $x = \dots\dots$
(b)
Work out angle $O C A$ . You must give a reason for each stage of your working.
[3 marks]
Answer
Total for D3: 6 marks
D4
$P A$ and $P B$ are tangents to the circle. $P A = (3 x + 1)$ cm and $P B = (5 x - 7)$ cm. Work out the length of $P A$ .
[3 marks]
Not drawn accurately
Answer
D5
A circle has radius 17 cm. A chord $A B$ is 30 cm long. A second chord $C D$ is parallel to $A B$ , on the other side of the centre, and 23 cm from $A B$ . Work out the length of $C D$ .
[4 marks]
Answer
D6
$A B$ is a diameter of the circle with centre $O$ , and $C$ is a point on the circle. Prove that angle $A C B = 9 0^{\circ}$ .
[4 marks]
Not drawn accurately
D7
$A$ , $B$ and $C$ are points on the circle with centre $O$ . $B$ is on the minor arc $A C$ and angle $A B C = 11 8^{\circ}$ . Work out the reflex angle $A O C$ . Give a reason.
[3 marks]
Not drawn accurately
Answer
D8
$P A$ is a tangent to the circle with centre $O$ , touching it at $A$ . Angle $O P A = 2 6^{\circ}$ . Work out angle $A O P$ . Give a reason.
[2 marks]
Answer
D9
$A B C D$ is a cyclic quadrilateral. Angle $B A C = 3 5^{\circ}$ and angle $C A D = 4 2^{\circ}$ . Work out angle $B C D$ . Give a reason.
[3 marks]
Not drawn accurately
Answer
D10
$A B C D$ is a cyclic quadrilateral. Angle $D A B = 2 x + 1 0^{\circ}$ , angle $B C D = 3 x - 5^{\circ}$ and angle $A B C = 4 x - 2 0^{\circ}$ . Work out the size of angle $A D C$ .
[4 marks]
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 tangents to a circle at $A$ and at $B$ meet at $T$ . $C$ is a point on the major arc $A B$ and angle $A C B = 5 7^{\circ}$ . Work out angle $A T B$ .
[3 marks]
Hint 1 · What to try
Join $A$ and $B$ to the centre $O$ . Two of the angles in quadrilateral $O A T B$ are already known.
Hint 2 · The first line
Angle $A O B = 2 \times 57 = 11 4^{\circ}$ , and angles $O A T$ and $O B T$ are both $9 0^{\circ}$ .
Hint 3 · The full method
The angles of $O A T B$ add up to $36 0^{\circ}$ , so angle $A T B = 360 - 90 - 90 - 114 = 6 6^{\circ}$ .
Answer

E2
A chord of a circle is 10 cm long. The shortest distance from the midpoint of the chord to the circle is 1 cm. Work out the radius of the circle.
[4 marks]
Hint 1 · What to try
Call the radius $r$ . The distance from the centre to the chord is $r - 1$ .
Hint 2 · The first line
The perpendicular from the centre halves the chord, so $r^{2} = 5^{2} + (r - 1)^{2}$ .
Hint 3 · The full method
$r^{2} = 25 + r^{2} - 2 r + 1$ , so $2 r = 26$ and $r = 13$ .
Answer
E3
$T A$ is a tangent to the circle with centre $O$ . $A B$ is a chord and $C$ is a point on the major arc $A B$ . Prove that angle $T A B =$ angle $A C B$ .
[4 marks]
Not drawn accurately
Hint 1 · What to try
Join $O$ to $A$ and to $B$ . Call angle $T A B$ by a letter, say $x$ .
Hint 2 · The first line
Angle $O A T = 9 0^{\circ}$ , so angle $O A B = 9 0^{\circ} - x$ . Triangle $O A B$ is isosceles, so angle $O B A$ is also $9 0^{\circ} - x$ .
Hint 3 · The full method
Then angle $A O B = 180 - 2 (90 - x) = 2 x$ , and the angle at the circumference is half of that, so angle $A C B = x$ .
E4
$O$ is the centre of a circle and $A B$ is a chord. $M$ is the point on $A B$ where the line from $O$ meets $A B$ at $9 0^{\circ}$ . Prove that $A M = M B$ .
[3 marks]
Hint 1 · What to try
Join $O$ to $A$ and to $B$ , and look at triangles $O M A$ and $O M B$ .
Hint 2 · The first line
$O A = O B$ (radii), $O M$ is shared, and both triangles have a right angle at $M$ .
Hint 3 · The full method
So the triangles are congruent (RHS), and $A M = M B$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the angle at the centre, the angle in a semicircle, angles in the same segment and cyclic quadrilaterals to find angles, giving reasons.
I can use the tangent and radius right angle, equal tangents, the perpendicular from the centre to a chord, and the alternate segment theorem.

Answers and mark scheme

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