question_answer

A circle is inscribed in an equilateral triangle of side a. What is the area of any square inscribed in this circle?

A)  \[\frac{{{a}^{2}}}{3}\]          

B)  \[\frac{{{a}^{2}}}{4}\]

C)  \[\frac{{{a}^{2}}}{6}\]

D)  \[\frac{{{a}^{2}}}{8}\]

Correct Answer: C

Solution :

Length of an equilateral triangle is \[a\frac{\sqrt{3}}{2}.\]

\[\therefore \]      Radius of incircle \[=\frac{a\sqrt{3}}{2}\times \frac{1}{3}=\frac{a}{2\sqrt{3}}\]

\[\therefore \]      Diameter of incircle \[=2\left( \frac{a}{2\sqrt{3}} \right)=\frac{a}{\sqrt{3}}\]

Let side of a square be x.

\[\therefore \]      \[{{\left( \frac{a}{\sqrt{3}} \right)}^{2}}={{x}^{2}}+{{x}^{2}}\Rightarrow \frac{{{a}^{2}}}{3}=2{{x}^{2}}\]

\[\therefore \]      \[{{x}^{2}}=\frac{{{a}^{2}}}{6}=\]Area of square