Circles

Why This Chapter Matters

A circle is highly symmetric. Every point on it is the same distance from the centre, and that symmetry creates strong relationships among chords, angles and distances.

The chapter develops in layers. First connect equal chords with equal central angles. Next use perpendiculars from the centre to understand chord length and distance. Then move from lengths to angles: an arc controls the angle at the centre and the angle at the circumference. Finally combine these ideas in cyclic quadrilaterals.

The most useful habit is to identify the same chord or same arc. Many circle theorems become obvious once you see which part of the circle the angles are standing on.

Bridge from Earlier Classes

Before starting, be comfortable with:

• Circle terminology
• Triangle congruence
• Isosceles triangles
• Perpendicular lines
• Angle sum

These are the tools this chapter reuses and deepens.

1. Chords and Central Angles

A chord joins two points on a circle. Equal chords subtend equal angles at the centre.

The proof uses congruent triangles formed by joining the chord endpoints to the centre. All radii are equal, and equal chords complete an SSS congruence.

The converse is also true: equal angles at the centre cut off equal chords.

2. Perpendicular from Centre to Chord

A perpendicular drawn from the centre of a circle to a chord bisects the chord.

The proof uses right triangles: the radii are equal, the perpendicular is common, and both triangles contain right angles, so RHS congruence applies.

Conversely, a line from the centre to the midpoint of a chord is perpendicular to that chord.

3. Equal Chords and Distance from Centre

Equal chords are equidistant from the centre, and chords at equal perpendicular distances from the centre are equal.

This explains the visual pattern that longer chords lie closer to the centre. The diameter, being the longest chord, passes through the centre and therefore has zero perpendicular distance from it.

4. Angle at Centre and Angle at Circle

For the same arc, the angle subtended at the centre is twice the angle subtended at a point on the remaining part of the circle.

This theorem generates several other results. If a central angle is 120°, an angle subtending the same arc at the circumference is 60°.

5. Same Segment and Semicircle

Angles subtended by the same chord in the same segment are equal.

A diameter subtends 180° at the centre, so any angle it subtends at the circumference is half of 180°, that is 90°. Hence the angle in a semicircle is a right angle.

6. Cyclic Quadrilaterals

A quadrilateral is cyclic when all four vertices lie on one circle.

Opposite angles of a cyclic quadrilateral are supplementary. Conversely, if a pair of opposite angles of a quadrilateral sum to 180°, the quadrilateral is cyclic.

This theorem turns a circle question into a simple angle-sum test.

7. Solving Circle Problems

Mark the centre, identify the relevant chord or arc, and look for isosceles triangles formed by radii.

If an angle is at the centre, think 'double'. If two angles stand on the same chord in the same segment, think 'equal'. If four points lie on a circle, think 'opposite angles supplementary'.

Visualise the Core Idea

Most Class 9 circle theorems come from symmetry around the centre and repeated use of congruent triangles.

Formula / Theorem Sheet

Worked Examples — Follow the Reasoning

How to Think in the Exam

Common Mistakes

Quick Revision

Focused Practice

Final Chapter Summary