JEE Main 2017 (Online) 9th AprilQuestion 1
The lengths of two adjacent sides of a cyclic quadrilateral are $2$ units and $5$ units and the angle between them is $60^\circ$. If the area of the quadrilateral is $4\sqrt3$ sq. units, then the perimeter of the quadrilateral is
Trigonometric Ratio & Identities
Solution
Concept. In a cyclic quadrilateral opposite angles add to $180^\circ$; split it along the diagonal opposite the $60^\circ$ angle. Formulas used. (1) Cosine rule; area $=\frac12xy\sin\theta$Step 1 - Diagonal. $d^2=4+25-2\cdot2\cdot5\cos60^\circ=19$. Step 2 - Areas. The first triangle has area $\frac12\cdot2\cdot5\sin60^\circ=\frac{5\sqrt3}{2}$, so the other has $4\sqrt3-\frac{5\sqrt3}{2}=\frac{3\sqrt3}{2}$. With sides $x,y$ and angle $120^\circ$: $\frac{\sqrt3}{4}xy=\frac{3\sqrt3}{2}$, so $xy=6$. Step 3 - $x+y$. Cosine rule with $120^\circ$: $x^2+y^2+xy=19$, so $(x+y)^2=19+6=25$ and $x+y=5$. Step 4 - Perimeter. $2+5+5=12$. Answer: (D)