Question

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

• A. (π/3√3)1/2
• B. (π/4)1/2
• C. (π/6)1/2
• D. (π/4√3)1/2

Answer 1

Answer: (A)

(π/3√3)1/2

Answer 2

The correct answer is (A):
It is given that radius of the circle = 1 cm
Chord AB subtends an angle of 60° on the centre of the given circle. R be the region bounded by the radii OA, OB and the arc AB.
Therefore, R = 60°/360°×Area of the circle = 1/6×π×(1)2 = π/6 sq.cm
It is given that OC = OD and area of triangle OCD is half that of R. Let OC = OD = x.
Area of triangle COD = 1/2×OC×OD×sin60°
π/6×2 = 1/2×x×x×√3/2
⇒ x2 = π/3√3
⇒ x = (π/3√3)1/2 cm.