Area of smaller region formed by chord

Question

A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of $120^\circ$ at the center. Then, the area, in square meters, of the smaller region is

Accepted Answer

C. Length of chord subtending angle $	heta = 120^\circ$ at center is $2R \sin\left(\frac{	heta}{2}ight) = 2R \sin(60^\circ) = R\sqrt{3}$. Given chord length = $10\sqrt{3} \implies R\sqrt{3} = 10\sqrt{3} \implies R = 10$. Area of smaller region = Area of sector - Area of triangle formed by chord and radii: $$	ext{Area of sector} = \frac{120^\circ}{360^\circ} \pi R^2 = \frac{1}{3} \pi (100) = \frac{100\pi}{3}$$ $$	ext{Area of triangle} = \frac{1}{2} R^2 \sin(120^\circ) = \frac{1}{2} (100) \left(\frac{\sqrt{3}}{2}ight) = 25\sqrt{3}$$ $$	ext{Area of smaller region} = \frac{100\pi}{3} - 25\sqrt{3} = 25\left(\frac{4\pi}{3} - \sqrt{3}ight)$$

Answer

$25\left(\frac{4\pi}{3} + \sqrt{3}\right)$

Answer

$20\left(\frac{4\pi}{3} - \sqrt{3}\right)$

Answer

$25\left(\frac{4\pi}{3} - \sqrt{3}\right)$

Answer

$20\left(\frac{4\pi}{3} + \sqrt{3}\right)$

Area of smaller region formed by chord

Explanation

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