Geometry · Topic 5 of 7

⭕ Circles

4 exam question types, fully solved

A circle turns a handful of theorems into a points machine: once you spot which arc, chord or tangent an angle is built on, the answer drops out in one line. SSC leans heavily on these laws because they reward recognition over calculation — learn the rules and most figures solve themselves.

The circle theorems that crack the section

• Angle at the centre: the angle an arc makes at the centre is twice the angle it makes at the circumference on the same arc — .

• Same segment: angles in the same segment (standing on the same chord) are equal.

• Angle in a semicircle: a diameter subtends at the centre, so any point on the circle sees it at — a right angle.

• Cyclic quadrilateral: opposite angles add to (not equal — that is a parallelogram).

• Perpendicular to a chord: the perpendicular from the centre bisects the chord, so . Equal chords are equidistant from the centre.

• Tangent ⊥ radius: a tangent is perpendicular to the radius at the point of contact, giving the right triangle .

• Two tangents: the two tangents drawn from an external point are equal in length.

• Alternate segment theorem: the angle between a tangent and a chord equals the inscribed angle in the alternate segment — they are simply equal, no doubling needed.

⭕Circle Theorem ExplorerPick a theorem — slide a point, watch the angles

Angle at the centre — the angle an arc makes at the centre is twice the angle it makes at the circumference on the same arc: ∠AOB = 2∠ACB.

Move C along the major arc

∠AOB = 140° = 2 × 70° = 2 ∠ACB

💡 Slide C — the triangle changes shape but ∠ACB stays 70°. Same chord, same segment ⇒ same angle. That is exactly the next theorem.

Using : if a central angle is , the inscribed angle on the same arc is °.

The same right-triangle idea links everything below: a chord and its distance from the centre, a tangent and the radius, even two circles and their common tangents all reduce to one step.

Q1O is the centre of a circle. If the central angle ∠AOB = 100°, the inscribed angle ∠ACB on the same arc is:

Q2A chord of length 24 cm is 5 cm from the centre of a circle. The radius is:

Q3The length of the tangent drawn from a point 13 cm from the centre of a circle of radius 5 cm is:

📚

Real exam questions — Circles

4 question types · 20 solved examples from real SSC papers

Every circle question is one theorem in disguise: centre angle is twice the inscribed angle, a radius bisects a chord it meets at right angles, and a tangent is perpendicular to the radius.

How to solve this type

Four laws cover almost every angle question.
(1) Angle at the centre

=2\times

angle at the circumference standing on the same arc:

\angle AOB = 2\,\angle ACB

.
(2) Angles in the same segment are equal.
(3) Angle in a semicircle

=90^\circ

(one side is a diameter).
(4) Opposite angles of a cyclic quadrilateral add to

180^\circ

.
Spot which arc or segment the angle stands on, then double, halve or subtract from

180^\circ

as needed.

O is the centre of a circle and $\angle AOB = 80^\circ$ . What is the inscribed angle $\angle ACB$ standing on the same arc AB?

160^\circ

80^\circ

20^\circ

40^\circ

An inscribed angle is $55^\circ$ . The central angle standing on the same arc is:

27.5^\circ

55^\circ

165^\circ

110^\circ

MN is a diameter of a circle and P is a point on the circle. What is $\angle MPN$ ?

60^\circ

180^\circ

90^\circ

45^\circ

In a cyclic quadrilateral ABCD, $\angle B = 82^\circ$ . Find $\angle D$ .

90^\circ

82^\circ

108^\circ

98^\circ

Two opposite angles of a cyclic quadrilateral are $(2x+10)^\circ$ and $(3x-20)^\circ$ . Find $x$ .

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