Question

A circle is inscribed in an equilateral triangle of side a. What is the area of any square inscribed in this circle?

(a) \(\frac{a^2}{3}\)

(b) \(\frac{a^2}{4}\)

(c) \(\frac{a^2}{6}\)

(d) \(\frac{a^2}{8}\)

Answer 1

(c) \(\frac{a^2}{6}.\)

If ‘a’ is length of the side of ΔABC, then

Area of ΔABC = \(\frac{\sqrt3}{4}\,a^2\)

semi-perimeter of ΔABC = \(\frac{3a}{2}\)

∴ Radius of in-circle = \(\frac{\text{Area}}{\text{semi-perimeter}}\) = \(\frac{\sqrt3}{4}\,a^2\) x \(\frac{2}{3a}\) = \(\frac{a}{2\sqrt3}\)

∴ Diagonal of square PQRS = Diameter of incircle.

= 2 x \(\frac{a}{2\sqrt3}\) = \(\frac{a}{\sqrt3}\)

 ∴ Area of squre = \(\frac{(\text{diagonal})^2}{2}\) = \(\frac{\big(\frac{a}{\sqrt3}\big)^2}{2}\) = \(\frac{a^2}{6}.\)