Rule
A perpendicular drawn from the centre of a circle to a chord bisects the chord (and conversely, a line from the centre that bisects a chord is perpendicular to it).
Equal chords of a circle (or of congruent circles) are equidistant from the centre — and the converse also holds.
Example 1 — radius from a chord's distance
A chord cm long is cm from the centre. Find the radius.
- The perpendicular from the centre bisects the chord: half-chord cm.
- This half-chord, the distance, and the radius form a right triangle: .
Answer: cm
Rule
Exactly one circle can pass through three non-collinear points — found as the intersection of the perpendicular bisectors of any two chords joining the points.
Rule
Equal arcs of a circle subtend equal angles at the centre — and equal central angles cut off equal arcs (the converse).
Example 2 — equal arcs, equal chords
If on a circle, what can be said about chords ?
- Equal arcs subtend equal central angles.
- Equal central angles cut off equal chords (same reasoning as the chord–distance theorems, run in reverse).
Answer: All four chords are equal in length
Quick Recap
- ●Perpendicular from centre to a chord bisects it, and conversely
- ●Equal chords equidistant from the centre
- ●Exactly one circle passes through three non-collinear points
- ●Equal arcs equal central angles equal chords
