Question

A circle is inscribed in an equilateral triangle of side $a$. The area of any square inscribed in the circle is $\frac{a^2}{k}$, where $k$ is

Answer 1

If $p$ be the altitude, then $p=a\sin {60}^0=\frac{a}{2}\sqrt{3}.$

Since the traingle is equilateral, therefore centroid, orthocentre, circumcentre all coincide.

Hence,radius of the in scribed 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

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

$\therefore$ Area $=x^2=\frac{a^2}{6}.$