Circles (NCERT Ch. 9)

• Define circle, radius, diameter, chord, arc, sector.
• Prove equal chords subtend equal angles at centre and converses.
• Use perpendicular from centre to chord bisects the chord.
• Understand angle subtended by arc at centre vs at point on circle.
• Apply cyclic quadrilateral angle property (intro).

1. Circle — Set of points equidistant from centre O; radius r.
2. Chord — Segment joining two points on circle.
3. Equal chords → equal central angles; equidistant from centre.
4. Perpendicular from centre to chord bisects chord.
5. Angle at centre = 2 × angle at circumference on same arc (major/minor care).
6. Arc — Portion of circumference; sector — region bounded by two radii and arc.
7. Cyclic quadrilateral — Opposite angles supplementary.
8. NCERT constructions link to this chapter.

Equal chords of a circle are equidistant from the centre — outline proof idea.

Use congruence of right triangles formed by radii to chord mid-points.
Perpendicular distance from O to chord is same for equal chords.

• Skipping given-to-prove structure in geometry proofs.
• Using wrong congruence criterion (SSA is not valid).
• Forgetting units in length/angle statements.

• State one NCERT result from Circles.
• Draw a neat diagram for a typical Circles problem.

Open the Practice tab for graded questions on Circles (NCERT Ch. 9).