Question

The length of two adjacent sides of a cyclic quadrilateral are $2$ and $5$ and the angle between them is ${60}^{\circ }.$ If length of the third side is $3,$ then length of the fourth side is

• A. $2$
• B. $3$
• C. $5$
• D. $8$

Answer 1

The correct option is A $2$


In a cyclic quadrilateral, we know that, sum of opposite angles is ${180}^{\circ }.$
So, $\angle CDB={180}^{\circ }-{60}^{\circ }={120}^{\circ }$

Let the remaining side length be $x.$
Using cosine rule in $△ABC,$ we get
$\cos {60}^{\circ }=\frac{2^2+5^2-B{C}^2}{2\times 2\times 5}\Rightarrow B{C}^2=19\cdots \left(1\right)$
Using cosine rule in $△BCD,$ we get
$\cos {120}^{\circ }=\frac{3^2+x^2-B{C}^2}{2\times x\times 3}$
From equation $\left(1\right)$, we get
$-\frac{1}{2}=\frac{9+x^2-19}{6x}\Rightarrow x^2+3x-10=0\Rightarrow (x+5)(x-2)=0\therefore x=2$