Question

Among all cyclic quadrilaterals inscribed in a circle of radius $R$ with one of its angles equal to $120^{\circ }$ Consider the one w ith maximum possible area. Its area is

• A. $\sqrt{2}{R}^2$
• B. $\sqrt{3}{R}^2$
• C. $2R^2$
• D. $2\sqrt{3}{R}^2$

Answer 1

Answer: (B)

$ABCD$ is a cyclic quadrilateral
$\angle A=120^{\circ }$

$ABCD$ has maximum area possible
When $\angle B=\angle D=90^{\circ }$
$\therefore$ Area of quadrilateral
$=2\times \frac{1}{2}\times AD\times DC$
$=2\times \frac{1}{2}\times R\times \sqrt{3}R$
$[\because AD=R,DC=\sqrt{3}R]$
$=\sqrt{3}{R}^2$