If a chord of a circle subtends an angle of 60° at the circumference of the circle, then what is the ratio of the radius of the circle and the length of the chord respectively?

Correct — D. The angle at the centre is double the angle at the circumference, so the chord subtends 120° at the centre.

Central angle: 2 × 60° = 120°
Chord = 2r × sin(half the central angle) = 2r × sin 60°
Substitute sin 60° = √3⁄2: chord = r√3

Radius : chord = r : r√3 = 1 : √3 → option (d)

Why the others are wrong

• (a)
  1:2 — 1 : 2 makes the chord 2r, a diameter. A diameter subtends 90° at the circumference, not 60°.
• (b)
  3:4 — 3 : 4 puts the chord at 4r⁄3, which needs sin 60° = 2⁄3. But sin 60° = √3⁄2 ≈ 0.866, and the chord is r√3 ≈ 1.73r.
• (c)
  2 : √5 — 2 : √5 puts the chord at (√5⁄2)r ≈ 1.12r. A 60° angle at the circumference needs a chord of r√3 ≈ 1.73r.

Two circle facts do the work. The angle a chord subtends at the centre is double the angle it subtends at a point on the major arc.

The two radii and the chord form an isosceles triangle, and the perpendicular from the centre bisects both the chord and the central angle. So chord = 2r sin(θ⁄2), where θ is the central angle.

With θ = 120°, half the chord is r sin 60° = r√3⁄2, and the chord is r√3.

Key facts

• The angle a chord subtends at the centre is double the angle it subtends at a point on the major arc.
• A chord that subtends angle θ at the centre has length 2r sin(θ⁄2).
• A chord equal to the radius subtends 60° at the centre.
• A diameter subtends 90° at every other point on the circle.

Common traps

• Taking 60° as the central angle, which makes the chord equal to the radius and the ratio 1 : 1.
• Writing the ratio the wrong way round. The stem asks for radius to chord, so r√3 goes second.

No directly related past PYQ was found.