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. $2\,R^{2}$
• 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] $
$=\sqrt{3}R^{2}$