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

1. $\bigg(\dfrac{\pi}{3\sqrt 3} \bigg)^\frac{1}{2} \\$
2. $\bigg(\dfrac{\pi}{4} \bigg)^\frac{1}{2} \\$
3. $\bigg(\dfrac{\pi}{6} \bigg)^\frac{1}{2} \\$
4. $\bigg(\dfrac{\pi}{4\sqrt 3} \bigg)^\frac{1}{2}$

Given that, radius $= \text{OA} = \text{OB} = 1 \; \text{cm},$ and $\boxed{\text{OC} = \text{OD}}$

So, the $\triangle \text{OCD},$ is isosceles triangle.

An isosceles triangle is a triangle that :

• Have two sides equal
• The base angles are also equal
• The perpendicular from the apex angle bisects the base

We can draw the diagram,

We know that, sum of all the angles of a triangle $= 180^ {\circ}$

Now, the sum of all the angles of a $\triangle \text{OCD} = 180^{\circ}$

$\Rightarrow 60^{\circ} + x + x = 180^{\circ}$

$\Rightarrow 2x = 120^{\circ}$

$\Rightarrow x = 60^{\circ}$

So, $\triangle \text{OCD}$ is a equilateral triangle.

Area of $\triangle \text{OCD} = \frac{1}{2} \text{(area of R)} \quad \longrightarrow (1)$

Area of sector $= \frac{Q}{360^{\circ}} \times \pi \times (\text{radius})^{2};$  where $Q$ is the angle subtended at the center.

Area of $R = \frac{60^{\circ}}{360^{\circ}} \times \pi \times (1)^{2} = \frac{\pi}{6} \; \text{cm}^{2}$

Now, the area of $\triangle \text{OCD} = \frac{\sqrt{3}}{4} \; \text{(side)}^{2} = \frac{\sqrt{3}}{4} \; \text{OC}^{2} \; \text{cm}^{2}$

From the equation $(1),$ we get

$\frac{\sqrt{3}}{4} \; \text{OC}^{2} = \frac{1}{2} \times \frac{\pi}{6}$

$\Rightarrow \text{(OC)}^{2} = \frac{\pi}{3 \sqrt{3}}$

$\Rightarrow \text{OC} = \sqrt{\frac{\pi}{3 \sqrt{3}}} = \left( \frac{\pi}{3 \sqrt{3}} \right)^{\frac{1}{2}} \; \text{cm}.$

Correct Answer $: \text{A}$

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