From a circle of radius 7 units, an arc length is cut by a chord of length 7 units. What is the arc length of the smaller portion (in units)?

Correct — D. The chord equals the radius, so the triangle it makes with the two radii is equilateral.

Chord = radius = 7, so all three sides are equal
Central angle = 60° = π⁄3 radians

Arc = r × θ, with θ in radians
= 7 × π⁄3
= 7π⁄3 → option (d).

Why the others are wrong

• (a)
  Option (a) is 7π⁄6, which is r × θ with θ = 30°. A chord equal to the radius fixes the angle at 60°, not half of it.
• (b)
  Option (b) is 7π⁄4, the arc of a 45° sector. Nothing in the data produces 45° — the equilateral triangle is what sets the angle.
• (c)
  Option (c) is 8π⁄3: the angle is right but the radius has slipped to 8. With r = 7 the sixth of the circumference is 7π⁄3.

Arc length is r × θ with θ in radians. That formula is not what decides this item — the angle is.

A chord equal in length to the radius forms an equilateral triangle with the two radii drawn to its ends, so the angle it subtends at the centre is 60°, that is π⁄3.

The smaller (minor) arc is the one facing that 60° angle, and 60° is one-sixth of a full turn. The major arc would be 7 × (2π − π⁄3) = 35π⁄3.

Key facts

• Arc length = rθ with θ measured in radians, and 60° = π⁄3.
• A chord equal to the radius subtends 60° at the centre, because chord and radii form an equilateral triangle.
• For radius 7 the full circumference is 14π, and one-sixth of it is 7π⁄3.

Common traps

• Substituting 60 rather than π⁄3 into rθ
• Giving the major arc when the question asks for the smaller portion

No directly related past PYQ was found.