Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $a=5,b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}.$

The side $c$ is?

• A. $6$
• B. $3$
• C. $2$
• D. none of these

Answer 1

The correct option is A $6$

we know that ,

$cosC=\frac{1-ta{n}^2\frac{C}{2}}{1+ta{n}^2\frac{C}{2}}=\frac{1-\frac{7}{9}}{1+\frac{7}{9}}=\frac{1}{8}$

Now, from cosine rule

$\frac{1}{8}=\frac{a^2+b^2-c^2}{2ab}\Rightarrow \frac{1}{8}=\frac{25+16-c^2}{2\times 4\times 5}\Rightarrow 5=25+16-c^2\Rightarrow c^2=36\Rightarrow c=6$

Therefore, Answer is $A$