question_answer

ABC is an equilateral triangle of side 4 cm. with A, B, C as vertex and radius 2 cm three arcs are drawn. The area of the region within the triangle bounded by the three area is

A)  \[\left( 3\sqrt{3}-\frac{\pi }{2} \right)\]\[c{{m}^{2}}\]     

B)  \[\left( \sqrt{3}-\frac{3\pi }{2} \right)\]\[c{{m}^{2}}\]

C)  \[4\left( \sqrt{3}-\frac{\pi }{2} \right)\]\[c{{m}^{2}}\]

D)  \[\left( \frac{\pi }{2}-\sqrt{3} \right)\]\[c{{m}^{2}}\]

Correct Answer: C

Solution :

(c): Each angle of the triangle = \[{{60}^{{}^\circ }}\] Required area of the three sectors \[=3\times \frac{60}{360}\times \pi {{(2)}^{2}}=2\pi c{{m}^{2}}\] Area of triangle \[=\frac{\sqrt{3}}{4}\times 16=4\sqrt{3}c{{m}^{2}}\] \[\therefore \]Required Area \[=(4\sqrt{3}-2\pi )c{{m}^{2}}\]