Chapter at a glance
The geometry of circles: symmetries, circles through given points, chords and the angles they subtend, perpendicular bisectors, arcs, and concyclic points.
What this chapter covers
- 5.1 Definitions
- 5.2 Symmetries of a Circle
- 5.3 How Many Circles?
- 5.4 Chords and the Angles They Subtend
- 5.5 Midpoints and Perpendicular Bisectors of Chords
- 5.6 Distance of Chords from the Centre
- 5.7 Angles Subtended by an Arc
- 5.8 Concyclicity of Points
Key points
- A circle has reflection symmetry in every diameter and rotational symmetry about its centre.
- Exactly one circle (the circumcircle) passes through three non-collinear points; its centre is where the perpendicular bisectors meet.
- Equal chords subtend equal angles at the centre and are equidistant from it (and conversely); the longer chord is nearer the centre.
- The perpendicular from the centre bisects a chord, and the line from the centre to a chord’s midpoint is perpendicular to it.
- The angle an arc subtends at the centre is twice the angle at any point on the rest of the circle; the angle in a semicircle is 90°.
Read the NCERT chapter
Read this chapter in the official NCERT textbook Ganita Manjari (2026-27 edition) for free on ncert.nic.in.
Source: NCERT, Ganita Manjari, Grade 9 (2026-27). NCERT chapter PDF. Spotted a mistake? Email edura.class9.yt@gmail.com. Last updated 11 October 2026.