question_answer

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is \[{{60}^{o}}\]. If the third side is 3, the remaining fourth side is [MNR 1994]

A) 2

B) 3

C) 4

D) 5

Correct Answer: A

Solution :

Let the fourth side be of  ?d? length. We know that \[A{{B}^{2}}+B{{C}^{2}}-2AB.BC\cos {{60}^{o}}=A{{C}^{2}}\] \[=C{{D}^{2}}+D{{A}^{2}}-2CD.DA\,\,\cos {{120}^{o}}\]     (by cosine rule) or \[4+25-2.\,2.\,5.\frac{1}{2}=9+{{d}^{2}}+3d\] Þ \[{{d}^{2}}+3d-10=0\Rightarrow d=-5\]or \[d=2\]; \ \[d=2\].