Multiple choice

In a circle with center O and radius 1 cm, an arc AB makes a 60 degree angle at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two respective points on OA and OB, 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 4 3 ) 1 2

  4. ( π 6 ) 1 2
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A Correct answer
Explanation

The sector has area (60/360)π = π/6. If OC = OD = x, triangle OCD has area (1/2)x^2 sin(60°) = sqrt(3)x^2/4. Equating this to half the sector area gives x = sqrt(π/(3sqrt(3))), corresponding to option A.

AI explanation

Using the sector area formula, the area of region R is (60 divided by 360) times pi times 1 squared, which simplifies to pi divided by 6. The area of triangle OCD is given as half of this, so it equals pi divided by 12. Since the included angle is 60 degrees, the area of triangle OCD is also one half times OC times OD times sine of 60 degrees. Substituting OC equals OD equals s, we get one half times s squared times the square root of 3 divided by 2 equals pi divided by 12. Solving for s yields s squared equals pi divided by (3 times the square root of 3), so the length OC is the square root of (pi divided by (3 times the square root of 3)).

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