GCSE · Circles, arcs & sectors

Tangents, chords & the alternate segmentExam-style questions

Why a tangent meets a radius at 90°, why two tangents from a point are equal, why the perpendicular from the centre bisects a chord, and the alternate segment theorem.

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13 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
$A$ , $B$ and $C$ are points on a circle with centre $O$ .

$TA$ and $TB$ are tangents to the circle.

Angle $ATB = 48°$ .Diagram NOT accurately drawn
(a) Work out the size of angle $AOB$ . Give reasons for your answer. (2 marks)
(b) Work out the size of angle $ACB$ . Give a reason for your answer. (2 marks)
Level 4 · 4 marksTry it
2
$A$ , $B$ and $C$ are points on a circle. $TA$ is a tangent to the circle at $A$ .

The angle between the tangent $TA$ and the chord $AB$ is $63°$ .

Angle $ABC = 71°$ .

(a) Write down the size of angle $ACB$ . Give a reason for your answer.

(b) Work out the size of angle $BAC$ .
Level 4 · 4 marksTry it
3
A circle has centre $O$ and radius $25$ cm. $AB$ is a chord of length $30$ cm.

(a) Work out the distance from $O$ to the chord $AB$ .

(b) $M$ is the midpoint of $AB$ , and the line $OM$ is extended to meet the circle at $N$ . Work out the length of $MN$ .
Level 4 · 5 marksTry it
4
$PT$ is a tangent to a circle with centre $O$ and radius $7$ cm, touching the circle at $T$ .

$PT = 24$ cm.Diagram NOT accurately drawn

(a) Work out the length $OP$ .

(b) Work out the shortest distance from $P$ to the circle.
Level 4 · 5 marksTry it
5
$PA$ and $PB$ are tangents to a circle with centre $O$ .

Angle $AOB = 138°$ .Diagram NOT accurately drawn

Work out the size of the angle marked $x$ . Give reasons for your answer.
Level 4 · 3 marksTry it