Circles

332. Equal Chords, Equal Angles · why matching chords always match their centre angles

The angles subtended by equal chords at the centre are always equal.

OBD∠AOB = 69°∠AOB = 69°∠COD = 69°∠COD = 69°AB = 225.9AB = 225.9CD = 225.9CD = 225.9AC
Equal chords subtend equal angles at the centre (and cut off equal arcs), and the converse holds too. Equal chords also lie equal distances from the centre — all reflections of the circle's symmetry.

Stuck? Ask Guru

Selina ICSE: Circles

What this lesson covers

Try to break it

Drag A and C around. Both chords stay equal in length, and the central angles ∠AOB and ∠COD always read the same value. Equal chords ⇔ equal central angles (by SSS: OA = OB = OC = OD = r, plus equal chords). Try to break the angle equality; impossible.

How you build it

Build two equal chords with their radii, and see ∠AOB = ∠COD.

  • Point tool: mark O near the middle — the centre of the circle.
  • Circle tool: click O as the centre, then click outward to set the radius.
  • Point tool: mark A on the circle.
  • Point tool: mark B elsewhere on the circle — AB is the first chord.
  • Segment tool: join A to B.
  • Point tool: mark C on the circle, somewhere away from A and B.
  • Compass tool: click A as the centre, then click B — this opens the compass to exactly the chord length AB and locks it.
  • Compass tool (still set to AB): click C — swing an arc that crosses the circle. That crossing is the same distance AB away from C.
  • Point tool: mark D where the arc crosses the circle. Now CD equals AB — a second, equal chord.
  • Segment tool: join C to D — the second chord, equal to AB.
  • Segment tool: join O to A.
  • Segment tool: join O to B. ∠AOB is the angle chord AB makes at the centre.
  • Segment tool: join O to C.
  • Segment tool: join O to D. ∠COD is the central angle of chord CD. Since CD = AB, the two equal chords subtend equal angles at the centre: ∠AOB = ∠COD.

The proof, step by step

Prove that equal chords subtend equal angles at the centre.

  • Given: AB = CD, two equal chords of a circle with centre O. To prove: ∠AOB = ∠COD.
  • In △AOB and △COD: OA = OC (radii of the same circle)
  • OB = OD (radii of the same circle)
  • AB = CD (given — the two chords are equal)
  • ∴ △AOB ≅ △COD (by the SSS congruence rule — all three pairs of sides are equal)
  • ∴ ∠AOB = ∠COD (c.p.c.t. — corresponding parts of congruent triangles). Hence, equal chords subtend equal angles at the centre. ∎

Worked example

In a circle with centre O, two chords AB and CD are equal in length. If ∠AOB = 70°, find the measure of ∠COD.

Equal chords of a circle subtend equal angles at the centre. Since AB = CD, ∠COD = ∠AOB = 70°.

  • 35°
  • 70° — correct
  • 110°
  • 140°
Hold to talk

Subscription Status