Question

A circle $C$ is drawn inside a square $S$ so that the four sides of $S$ are tangents to $C$. An equilateral triangle $T$ is drawn inside $C$ with its vertices on $C$. If the area of $S$ is $k$ times the area of $T$, then the value of $k$ is

• A. $\frac{16}{3\sqrt{3}}$
• B. $\frac{16}{\sqrt{3}}$
• C. $\frac{32}{3\sqrt{3}}$
• D. $\frac{32}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{16}{3\sqrt{3}}$

$a=2r;b=\sqrt{3}r$
$S=K.T$
$a^2=k.\frac{\sqrt{3}}{4}{b}^2$

$\Rightarrow 4r^2=k.\frac{\sqrt{3}}{4}{\left(\sqrt{3}r\right)}^2$

$\Rightarrow k=\frac{16}{3\sqrt{3}}$