question_answer

Let A be the area of a square inscribed in a circle of a radius r and let B be the area of a hexagon inscribed in the same circle. Then the ratio \[\frac{B}{A}\] equals

A)  \[2\sqrt{3}\]   

B)    \[\frac{3\sqrt{3}}{4}\]

C)  \[\frac{2\sqrt{3}}{4}\]          

D)    None of these

Correct Answer: B

Solution :

     Let r be the radius of the circle and x be the side of the square inscribed in it. Then, \[2{{x}^{2}}={{(2r)}^{2}}\,\Rightarrow \,2{{x}^{2}}=4{{r}^{2}}\] \[\Rightarrow \,\,{{x}^{2}}=2{{r}^{2}}=\] the area of the inscribed square Let ABCDEF be a hexagon inscribed in the same circle. Area of the hexagon \[=6\times \] Area of the equilateral triangle OBC \[=6\times \frac{\sqrt{3}}{4}\times {{r}^{2}}=\frac{3\sqrt{3}}{2}\,{{r}^{2}}\]   Required ratio \[=\frac{\frac{3\sqrt{3}{{r}^{2}}}{2}}{2{{r}^{2}}}\,=\frac{3\sqrt{3}}{4}=3\sqrt{3}:4\]