Question

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

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is A $2$


In $△ABD$
$\cos {60}^{\circ }=\frac{2^2+5^2-B{D}^2}{2\times 2\times 5}\Rightarrow \frac{1}{2}=\frac{29-B{D}^2}{2\times 2\times 5}\Rightarrow B{D}^2=19$

In $△BCD$
$\cos {120}^{\circ }=\frac{x^2+9-19}{2x\times 3}\Rightarrow \frac{-1}{2}=\frac{x^2-10}{6x}\Rightarrow x^2+3x-10=0\Rightarrow (x+5)(x-2)=0\Rightarrow x=2or-5\text{(Not Possible)}$
$\therefore \text{Length of the fourth side is}=2$