Question

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

• A. $12.5$ units
• B. $13$ units
• C. $13.2$ units
• D. $12$ units

Answer 1

Answer: (D)

$cos60^o=\frac{4+25-c^2}{\mathrm{2.2.5}}$
$\Rightarrow 10=29-c^2$
$\Rightarrow c^2=19$
$\Rightarrow c=\sqrt{19}$
Now,
$cos120°=\frac{a^2+b^2-c^2}{2ab}$
$\Rightarrow$ $-\frac{1}{2}=\frac{a^2+b^2-19}{2ab}$
$\Rightarrow a^2+b^2-19=-ab$
$\Rightarrow a^2+b^2+ab=19$
Area of quadrilateral $=\frac{1}{2}\times 2\times 5sin60°+\frac{1}{2}absin120°=4\sqrt{3}$
$\Rightarrow \frac{5\sqrt{3}}{2}+\frac{ab\sqrt{3}}{4}=4\sqrt{3}$
$\Rightarrow \frac{ab}{4}=4-\frac{5}{2}=\frac{3}{2}$
$\Rightarrow ab=6$
$\Rightarrow a^2+b^2=13$
$\Rightarrow a=2,b=3$
Perimeter
$=2+5+2+3$
$=12$