Question

The two adjacent sides of a cyclic quadrilateral are $2$ and $5$ and the angle between them is $60^{\circ }$. If the area of the quadrilateral is $4\sqrt{3}$ , then the perimeter of the quadrilateral is :

• A. 12.5
• B. 13.2
• C. 12
• D. 13

Answer 1

Answer: (B)

$\cos 60=\frac{4+25-C^2}{\mathrm{2.2.5}}$
$10=29-C^2$
$C^2=19$
$C=\sqrt{19}$
$-\frac{1}{2}=\frac{a^2+b^2-19}{2ab}$
$a^2+b^2-19=-ab$
$a^2+b^2+ab=19$
Area $=\frac{1}{2}\times 2\times 5\sin 60+\frac{1}{2}ab\sin x=4\sqrt{3}$
$\frac{5\sqrt{3}}{2}+\frac{ab\sqrt{3}}{4}=4\sqrt{3}$
$\frac{ab}{4}=4-\frac{5}{2}=\frac{3}{2}$
$ab=6$
$a^2+b^2=13$
$a=2,b=3$
Perimeter
$=2+5+2+3$
$=12$