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(D) The area of an equilateral triangle with side $a$ is $\Delta = \frac{\sqrt{3}}{4} a^2$. The semi-perimeter of the triangle is $s = \frac{3a}{2}$.

Vedclass · Accepted Answer

$\frac{a^2}{6}$

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(D) The area of an equilateral triangle with side $a$ is $\Delta = \frac{\sqrt{3}}{4} a^2$. The semi-perimeter of the triangle is $s = \frac{3a}{2}$. The radius $r$ of the incircle is given by $r = \frac{\Delta}{s} = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{3a}{2}} = \frac{a}{2 \sqrt{3}}$. $A$ square inscribed in a circle of radius $r$ has a diagonal equal to the diameter of the circle,which is $2r$. Diagonal of the square $= 2 	imes \frac{a}{2 \sqrt{3}} = \frac{a}{\sqrt{3}}$. The area of a square with diagonal $d$ is $\frac{d^2}{2}$. Area of the square $= \frac{(\frac{a}{\sqrt{3}})^2}{2} = \frac{\frac{a^2}{3}}{2} = \frac{a^2}{6}$.

If a circle is inscribed in an equilateral triangle of side $a$,then the area of any square (in sq. units) inscribed in this circle is

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