A chord of a circle has a length of 12 cm. The angle subtended by the chord at a point on the circumference is 30°. What is the distance from the center of the circle to the chord?

Correct — A. Turn the angle at the circumference into the angle at the centre.

Central angle = 2 × 30° = 60°
OA = OB = r with 60° between them, so triangle OAB is equilateral
r = chord = 12 cm

The perpendicular OM bisects the chord: AM = 6 cm
OM = √(12² − 6²) = √108 = 6√3 cm → option (a)

Why the others are wrong

• (b)
  3√3 cm — 3√3 is the height of an equilateral triangle of side 6, half the true one. Triangle OAB has side 12, because the radius equals the full chord, not the half-chord.
• (c)
  6 cm — 6 cm is the half-chord AM, the other leg of triangle OMA. OM equals AM only when ∠AOM is 45° (a 45° angle at the circumference); here ∠AOM is 30°.
• (d)
  3 cm — 3 cm is far too short: OM² = 12² − 6² = 108, so OM ≈ 10.4 cm. With OM = 3 the radius would be √45 ≈ 6.7 cm, and the chord would subtend about 63° at the circumference.

Two circle theorems chain here. A chord's angle at the centre is twice its angle at a point on the major arc, so 30° there means 60° at the centre.

The perpendicular from the centre to a chord bisects it, so the distance, the half-chord and the radius form a right triangle.

Key facts

• Angle at the centre = 2 × angle at a point on the major arc, for the same chord.
• A 60° central angle makes the chord equal to the radius.
• Distance from the centre to a chord = √(r² − (half-chord)²).

Common traps

• Treating 30° as the central angle — triangle OAB is then not equilateral and the radius is not 12 cm.
• Building the equilateral triangle on the half-chord (side 6) instead of the full chord (side 12), which halves the answer to 3√3.

No directly related past PYQ was found.