Exam

JEE Main

Year

2017

Date

9 April

Mode

Online

Subject

Mathematics

Chapter

Trigonometric Ratios & Identities

JEE Main 2017 · 9 April · Online · Mathematics · Trigonometric Ratios & Identities

The lengths of two adjacent sides of a cyclic quadrilateral are 2...

The lengths of two adjacent sides of a cyclic quadrilateral are $2$ units and $5$ units and the angle between them is ${60}^{o}$. If the area of the quadrilateral is $4\sqrt{3}$ sq. units, then the perimeter of the quadrilateral is

Step-by-step solution

$cos {60}^{o} = \dfrac{4 + 25 - {c}^{2}}{2.2 . 5}$

$⇒ 10 = 29 - {c}^{2}$

$⇒ {c}^{2} = 19$

$⇒ c = \sqrt{19}$

Now,

$\cos 120° = \dfrac{{a}^{2} + {b}^{2} - {c}^{2}}{2ab}$

$⇒$ $- \dfrac{1}{2} = \dfrac{{a}^{2} + {b}^{2} - 19}{2ab}$

$⇒ {a}^{2} + {b}^{2} - 19 = - ab$

$⇒ {a}^{2} + {b}^{2} + ab = 19$

Area of quadrilateral $= \dfrac{1}{2} \times 2 \times 5\sin 60° + \dfrac{1}{2}absin 120° = 4\sqrt{3}$

$⇒ \dfrac{5\sqrt{3}}{2} + \dfrac{ab\sqrt{3}}{4} = 4\sqrt{3}$

$⇒ \dfrac{ab}{4} = 4 - \dfrac{5}{2} = \dfrac{3}{2}$

$⇒ ab = 6$

$⇒ {a}^{2} + {b}^{2} = 13$

$⇒ a = 2, b = 3$

Perimeter

$= 2 + 5 + 2 + 3$

$= 12$ units