Question

In $\Delta A B C$, if $2 R+r=r_{2}$, then $\angle B$ is equal to

• A. $\frac{\pi}{3}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{6}$
• D. $\frac{\pi}{2}$

Answer 1

Answer: (D)

We have,
$ 2 R+r=r_{2} $
$\Rightarrow 2 R=r_{2}-r $
$\Rightarrow 2 R=4 R \sin \frac{B}{2} \cos \frac{A}{2} \cos \frac{C}{2}-4 R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$
$\Rightarrow 2 R=4 R \sin \frac{B}{2} \cos \left(\frac{A+C}{2}\right) $
$\Rightarrow 1=2 \sin ^{2} \frac{B}{2} $
$ \Rightarrow \sin ^{2} \frac{B}{2}=\frac{1}{2} $
$ \Rightarrow \sin \frac{B}{2}=\frac{1}{\sqrt{2}}=\sin \frac{\pi}{4} $
$\Rightarrow \frac{B}{2}=\frac{\pi}{4} $
$\Rightarrow B=\frac{\pi}{2}$