A chord AB of a circle has a length of 10 cm. The angle subtended by the chord at a point C on the major arc is 30°. Find the radius of the circle.

Correct — B. The angle at the centre is double the angle at the circumference on the same arc.

Central angle: ∠AOB = 2 × 30° = 60°
Triangle AOB: OA = OB = r, so each base angle is (180° − 60°) ÷ 2 = 60°
All three angles are 60°, so △AOB is equilateral

Radius = chord: r = AB = 10 cm → option (b)

Why the others are wrong

• (a)
  7 cm — 7 cm is shorter than the 10 cm chord, so the apex angle of △AOB would exceed 60°. The angle at C would then be about 46°, not 30°.
• (c)
  11 cm — 11 cm is longer than the chord, so the apex angle of △AOB falls below 60°. The angle at C would then be about 27°, not 30°.
• (d)
  17 cm — 17 cm leaves the chord far shorter than the radius. The central angle drops to about 34°, so the angle at C is about 17°, not 30°.

The central angle on an arc is double the angle the same arc subtends anywhere on the rest of the circle. C sits on the major arc, so ∠ACB stands on the minor arc AB and ∠AOB = 60°.

OA and OB are both radii. An isosceles triangle with a 60° apex has all three angles equal to 60°, so all three sides are equal and the radius equals the chord.

Key facts

• The angle at the centre is double the angle at the circumference standing on the same arc.
• A chord that subtends 60° at the centre equals the radius.
• Extended sine rule: chord = 2r × sin(angle at the circumference).

Common traps

• Treating 30° as the central angle instead of doubling it to 60°.
• Halving instead of doubling, 30° ÷ 2 = 15°, which reverses the centre-circumference rule.

No directly related past PYQ was found.