Question

A circle is inscribed in an equilateral triangle of side a, the area of any square inscribed in the circle is

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

If p be the altitude, then p = asin ${60}^{\circ }$ = $\frac{a}{2}\sqrt{3}$.

Since the triangle is equilateral, therefore centroid, orthocentre, circumcentre and incentre all coincide.

Hence, radius of the inscribed circle = $\frac{1}{3}p=\frac{a}{2\sqrt{3}}=r$ or $diameter=2r=\frac{a}{\sqrt{3}}$.

Now if x be the side of the square inscribed, then angle in a semicircle being a right angle, hence

$x^2+x^2=d^2=4r^2\Rightarrow 2x^2=\frac{a^2}{3}$