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 \(4\sqrt3\), then the perimeter of the quadrilateral is :
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 \(4\sqrt3\), then the perimeter of the quadrilateral is :
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Let the other two sides be \(c,d\). Diagonal opposite the \(60^\circ\) angle: \(\delta^2=4+25-2\cdot2\cdot5\cos60^\circ=19\).
The opposite angle is \(120^\circ\): \(\delta^2=c^2+d^2-2cd\cos120^\circ=c^2+d^2+cd=19\).
Area \(=\frac12(2)(5)\sin60^\circ+\frac12cd\sin120^\circ=\frac{5\sqrt3}{2}+\frac{\sqrt3}{4}cd=4\sqrt3\Rightarrow cd=6\).
So \(c^2+d^2=13\) and \((c+d)^2=13+12=25\Rightarrow c+d=5\) (sides \(2,3\)).
Perimeter \(=2+5+5=12\).
Answer: (3) 12
The opposite angle is \(120^\circ\): \(\delta^2=c^2+d^2-2cd\cos120^\circ=c^2+d^2+cd=19\).
Area \(=\frac12(2)(5)\sin60^\circ+\frac12cd\sin120^\circ=\frac{5\sqrt3}{2}+\frac{\sqrt3}{4}cd=4\sqrt3\Rightarrow cd=6\).
So \(c^2+d^2=13\) and \((c+d)^2=13+12=25\Rightarrow c+d=5\) (sides \(2,3\)).
Perimeter \(=2+5+5=12\).
Answer: (3) 12
Updated On: Nov 11, 2026
- \(12.5\)
- \(13.2\)
- \(12\)
- \(13\)
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The Correct Option is C
Solution and Explanation
Use the cosine rule on the diagonal for both triangles (opposite angles are supplementary) and the area to find the other two sides.
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