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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

Explanation:

Let OC = OD = x

△OCD is an equilateral triangle of side x. [All angles are 60 degree.]

A(△OCD) = $\frac{\sqrt{3}}{4} x^{2}$

A(sector O-AB) = $\frac{60}{360} \times π {(1)}^{2}$ = $\frac{π}{6}$

By the given condition,

$\frac{\sqrt{3}}{4} x^{2}$ = $\frac{1}{2} \times \frac{π}{6}$

∴ x² = $\frac{π}{3 \sqrt{3}}$

∴ x = ${(\frac{π}{3 \sqrt{3}})}^{1 / 2}$

Hence, option (b).

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