Question

The 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 \(4\sqrt{3}\) sq. units then the perimeter of quadrilateral (in units) is:

• A. \(14\)
• B. \(13\)
• C. \(12\)
• D. \(10\)

Hint:

Opposite angles are supplementary; use the cosine rule on the diagonal and the area formula to find the sum of the other two sides.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Let the sides be 2 and 5 with included angle \(60^{\circ}\); the opposite angle is \(120^{\circ}\). Let the other sides be \(x\) and \(y\).

Step 2: Use the diagonal
Diagonal squared \(=4+25-2\cdot 2\cdot 5\cos 60^{\circ}=19\).
In the other triangle: \(x^2+y^2-2xy\cos 120^{\circ}=19\), so \(x^2+y^2+xy=19\).

Step 3: Use the area
Area \(=\frac{1}{2}(2)(5)\sin 60^{\circ}+\frac{1}{2}xy\sin 120^{\circ}=\frac{5\sqrt3}{2}+\frac{\sqrt3}{4}xy=4\sqrt3\)
So \(xy=6\), and \(x^2+y^2=13\).

Step 4: Perimeter
\((x+y)^2=13+12=25\), so \(x+y=5\).
Perimeter \(=2+5+5=12\).

Final Answer:
The perimeter of the quadrilateral is 12 units. \[ \boxed{\text{(C) }12} \]