question_answer

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is \[{{60}^{\text{o}}}\] . If the area of the quadrilateral is \[4\sqrt{3},\] then the perimeter of the quadrilateral is: [JEE Online 09-04-2017]

A)  12.5                                      

B)  13

C)  13.2                                      

D)  12

Correct Answer: D

Solution :

                \[\cos 60\,=\frac{4+25\,-{{c}^{2}}}{2.2.5}\] \[10=29-{{c}^{2}}\] \[{{c}^{2}}=19\] \[\] \[-\frac{1}{2}\,=\frac{{{a}^{2}}+{{b}^{2}}-19}{2ab}\] \[{{a}^{2}}+{{b}^{2}}-19=-ab\] \[{{a}^{2}}+{{b}^{2}}+ab=19\] Area \[=\frac{1}{2}\,\times 2\times 5\,\sin 60\,+\frac{1}{2}\,ab\,\sin x\,=4\sqrt{3}\] \[\frac{5\sqrt{3}}{2}\,+\frac{ab\sqrt{3}}{4}\,=4\sqrt{3}\] \[\frac{ab}{4}\,=4-\frac{5}{2}=\frac{3}{2}\] \[\] \[{{a}^{2}}+{{b}^{2}}=13\] \[\] Perimeter \[=2+5+2+3\] \[=12\]