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
$TA$ and $TB$ are tangents to a circle with centre $O$ . $C$ is a point on the circle. Angle $ATB = 64°$ .Diagram NOT accurately drawn

(a) Work out the size of angle $AOB$ . Give reasons for your answer. [2]

(b) Work out the size of angle $ACB$ . Give a reason for your answer. [2]
Level 3 · 4 marksTry it
2
The diagram shows a circle with centre $O$ and radius $13$ cm. $AB$ is a chord of length $24$ cm.Diagram NOT accurately drawn

Work out the distance from the centre $O$ to the chord $AB$ .
Level 3 · 3 marksTry it
3
$PA$ and $PB$ are tangents to a circle with centre $O$ .

Angle $APB = 48°$ .Diagram NOT accurately drawn

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

(b) Work out the size of angle $AOB$ .
Level 3 · 4 marksTry it
4
$PA$ is a tangent to a circle with centre $O$ and radius $8$ cm. The point $A$ is where the tangent touches the circle.

$OP = 17$ cm.

Work out the length of $PA$ .
Level 3 · 3 marksTry it
5
$PA$ and $PB$ are tangents to a circle with centre $O$ , touching the circle at $A$ and at $B$ .

$PA = 9$ cm.Diagram NOT accurately drawn

(a) Write down the length of $PB$ . Give a reason for your answer.

(b) Write down the size of angle $OAP$ . Give a reason for your answer.
Level 3 · 4 marksTry it