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
$R$ and $S$ are points on a circle, centre $O$ .

The tangent to the circle at $R$ and the tangent to the circle at $S$ meet at the point $T$ .Prove that $RT = ST$

(4 marks)
Level 5 · 4 marksTry it
2
$A$ , $B$ and $C$ are points on a circle. $TA$ is a tangent to the circle at $A$ .

Angle $TAB = 58°$ and $CA = CB$ .Diagram NOT accurately drawn

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

(b) Work out the size of angle $CAB$ . Give a reason for your answer.
Level 5 · 4 marksTry it
3
A circle has centre $O$ . The chord $AB$ has length $16$ cm, and the perpendicular distance from $O$ to $AB$ is $6$ cm.Diagram NOT accurately drawn

(a) Work out the radius of the circle.

The point $P$ is $26$ cm from $O$ . $PT$ is a tangent to the circle, touching it at $T$ .

(b) Work out the length $PT$ .
Level 5 · 5 marksTry it