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)

$cos 60^{o}=\frac{4 + 25 - c^{2}}{2.2 . 5}$
$\Rightarrow 10=29-c^{2}$
$\Rightarrow c^{2}=19$
$\Rightarrow c=\sqrt{19}$
Now,
$cos 120 ° = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$
$\Rightarrow $ $-\frac{1}{2}=\frac{a^{2} + b^{2} - 19}{2 a b}$
$\Rightarrow a^{2}+b^{2}-19=-ab$
$\Rightarrow a^{2}+b^{2}+ab=19$
Area of quadrilateral $=\frac{1}{2}\times 2\times 5sin 60°+\frac{1}{2}absin ⁡ 120°=4\sqrt{3}$
$\Rightarrow \frac{5 \sqrt{3}}{2}+\frac{a b \sqrt{3}}{4}=4\sqrt{3}$
$\Rightarrow \frac{a b}{4}=4-\frac{5}{2}=\frac{3}{2}$
$\Rightarrow ab=6$
$\Rightarrow a^{2}+b^{2}=13$
$\Rightarrow a=2, \, b=3$
Perimeter
$=2+5+2+3$
$=12$