Question

A square and the an equilateral triangle are inscribed in a circle If $a$ and $b$ denote the length of their sides then

• A. $a^2=\frac{b^2}{2}$
• B. $\frac{a^2}{2}=b^2$
• C. $3b^2=2a^2$
• D. $3a^2=2b^2$

Answer 1

The correct option is D $3a^2=2b^2$

We have to express the radius of the circle in terms of a and b.
$a^2=r^2+r^2$
$=2r^2$
$\Rightarrow r^2=\frac{a^2}{2}$ ....(i)
$\frac{AC}{r}=\cos {30}^0\Rightarrow \frac{b}{2}.\frac{1}{r}=\frac{\sqrt{3}}{2}$
$\therefore r=\frac{b}{\sqrt{3}}$ or $\therefore r^2=\frac{b^2}{\sqrt{3}}$ ...(ii)
From (i) and (ii), we have
$\frac{a^2}{2}=\frac{b^2}{3}\Rightarrow 3a^2=2b^2$