Question

The sides of a triangle are in the ratio $5:8:11$. If $\theta$ denotes the angle opposite to the largest side of the triangle, then the value of ${\tan }^2\frac{\theta }{2}$ is equal to

• A. $\frac{1}{21}$
• B. $\frac{7}{48}$
• C. $\frac{7}{3}$
• D. $\frac{7}{12}$

Answer 1

The correct option is C

$\frac{7}{3}$

Explanation for the correct option:

Find the required value.

Given: the sides of a triangle are in the ratio $5:8:11$.

Let the sides be $5x,8x,$ and $11x$.

Then $s=\frac{(5x+8x+11x)}{2}$

$s=12x$

$\theta$ is opposite to the side $11x$.

$$\begin{gathered} {\tan }^2\frac{\theta }{2}=\frac{(s–5x)(s–8x)}{s(s–11x)} \\ {\tan }^2\frac{\theta }{2}=\frac{(12x–5x)(12x–8x)}{12x(12x–11x)} \\ {\tan }^2\frac{\theta }{2}=\frac{144x^2-96x^2-60x^2+40x^2}{144x^2-132x^2} \\ {\tan }^2\frac{\theta }{2}=\frac{28x^2}{12x^2} \\ {\tan }^2\frac{\theta }{2}=\frac{7}{3} \end{gathered}$$

Hence, option $C$ is the correct answer.