Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
The two adjacent sides of a cyclic quadrilateral are $2, 5$ and the angle between them is $60°$. If the area of the quadrilateral is $4\sqrt{3}$, then the remaining two sides are:
$1, 2$
$2, 2$
$2, 3$
None of these
Step-by-Step Solution
Key Concept: For cyclic quadrilaterals, opposite angles sum to $180°$, and the quadrilateral area equals the sum of two triangular areas.
Given $AB = 2$, $BC = 5$, and $\angle ABC = 60°$, with cyclic quadrilateral $ABCD$ we have $\angle CDA = 180° - 60° = 120°$. Using the area formula: Area($ABCD$) = Area($\triangle ABC$) + Area($\triangle ACD$) = $\frac{1}{2}\cdot 2\cdot 5\cdot\sin 60° + \frac{1}{2}\cdot c\cdot d\cdot\sin 120°$. Substituting $c = CD$ and $d = DA$ with $cd = 4\sqrt{3}$, the total area becomes $\frac{5\sqrt{3}}{2} + \sqrt{3} = 4\sqrt{3}$.
Correct Answer: 3