Mathematics · Class 9
Circles
Chords, arcs, angles subtended, and cyclic quadrilaterals.
3 sections15–25 min read6 MCQs · 2 examples
Chapter Overview
Circles are sets of points equidistant from a centre. Class 9 covers chords, arcs, segments, angles subtended by chords, and cyclic quadrilateral properties.
Basic Terms
- Radius: distance from centre to circle
- Diameter: longest chord, equals 2r
- Chord: line segment joining two points on circle
- Arc, sector, segment — parts of circle
Key Theorems
- Equal chords subtend equal angles at centre
- Perpendicular from centre to chord bisects the chord
- Equal chords are equidistant from centre
- Angle in semicircle is a right angle
- Cyclic quadrilateral: opposite angles supplementary
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Chord distance
A chord of length 8 cm is 3 cm from centre. Find radius.
- 1Perpendicular bisects chord → half chord = 4 cm
- 2r² = 3² + 4² = 25
- 3r = 5 cm
Answer
Radius = 5 cm
Example 2: Cyclic quadrilateral
In cyclic quadrilateral ABCD, ∠A = 85°. Find ∠C.
- 1Opposite angles supplementary
- 2∠C = 180° − 85°
Answer
∠C = 95°
Key Points to Remember
- ✓ Only one circle passes through three non-collinear points
- ✓ Equal chords → equal arcs
- ✓ Angle at centre is twice angle at circumference on same arc
Exam Tips
- • Draw radius to chord for right triangle
- • Label centre O clearly
- • Use cyclic quad property for angle problems
Practice MCQs — Circles
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Question 1 of 6Score: 0/0
Longest chord of circle is: