In a circle with center O and radius 1 cm

Geometry

In a circle with center O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points
on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

  1. (π/3√3)1/2
  2. (π/4)1/2
  3. (π/6)1/2
  4. (π/4√3)1/2

Answer

Radius of the circle = 1 cm

Chord AB subtends an angle of 60° on the centre of the circle. R is the region bounded by the radii OA, OB and the arc AB.

R = 60°/360° × Area of the circle

= 1/6 × π × (1)2

= π/6 sq. cm

Given that OC = OD and area of triangle OCD is half that of R.

Area of triangle OCD = 1/2 × OC × OD × sin 60°

π/6 × 1/2 = 1/2 × OC × OC × √3/2

⇒ OC2 = π/3√3 

The correct option is A.