Chords and angles at the centre · 2 marks

In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB?

Answer: OA = OB = 12 cm, so the base angles of ΔOAB are (180° − 60°) ÷ 2 = 60° each. The triangle is equilateral, so AB = 12 cm.

Step-by-step solution

Given: OA = OB = 12 cm (radius); ∠AOB = 60°
To find: Length of chord AB

Idea: The triangle formed by a chord and the centre is isosceles. When its angle at the centre is 60°, the other two angles are also 60°, so it is equilateral and the chord equals the radius.

In ΔOAB, OA = OB = 12 cm (radii), so ∠OAB = ∠OBA (angles opposite equal sides).½ mark
∠OAB + ∠OBA = 180° − 60° = 120°, so each is 60°.½ mark
All three angles are 60°, so ΔOAB is equilateral and AB = OA = 12 cm.1 mark

Chord AB = 12 cm (ΔOAB is equilateral).

Check: A regular hexagon inscribed in a circle has sides equal to the radius, and each side subtends 360° ÷ 6 = 60° at the centre ✓.

Answer to write in the exam

OA = OB = 12 cm (radii) ⇒ ∠OAB = ∠OBA

∠OAB + ∠OBA = 180° − 60° = 120° (angle sum of ΔOAB) ⇒ ∠OAB = ∠OBA = 60°

⇒ ΔOAB is equilateral

∴ AB = OA = 12 cm

Common mistakes that cost marks

Using the arc length formula or 60/360 of the circumference. The question asks for the straight chord, not the arc.
Answering 24 cm (the diameter) or 6 cm (half the radius).
Stopping at “isosceles”; the 60° angle is what makes the triangle equilateral.

How this can come in the exam

MCQ (1 mark)

A chord of a circle of radius 6 cm subtends a right angle at the centre. Its length is

6 cm
6√2 cm
12 cm
3√2 cm

Show answer

(B) 6√2 cm
By the Baudhāyana–Pythagoras theorem in the right isosceles triangle: chord = √(6² + 6²) = 6√2 cm.

Try one yourself

In a circle of radius 7.5 cm, chord PQ subtends 60° at the centre. Find PQ.