Question

A square $ABCD$ in inscribed in a circle of radius a. Another circle is inscribed in $ABCD$ and a square $EFGH$ is inscribed in this circle. The side $EF$ is equal to

• A. $a$
• B. $a\sqrt{2}$
• C. $\frac{a}{\sqrt{2}}$
• D. $\frac{a}{2}$

Answer 1

The correct option is A $a$

Let the side of square ABCD $=$ x cm and the side of the square EFGH $=$ y cm.

In right angled triangle OAE, $O{A}^2=O{E}^2+E{A}^2$
$⟹a^2={\left(\frac{x}{2}\right)}^2+{\left(\frac{x}{2}\right)}^2=\frac{x^2+x^2}{4}$
$\therefore 2a^2=x^2$
In right angled triangle EHG, $E{G}^2=E{H}^2+H{G}^2$
$⟹x^2=y^2+y^2$
$⟹2y^2=x^2=2a^2$
$\therefore y=a=EF$