Question

In a $\Delta ABC$, let $\angle C=\frac{\pi }{2}$, if $r$ is the inradius and $R$ is the circumradius of the $\Delta ABC$, then $2(r+R)$ is equal to

• A. $c+a$
• B. $a+b+c$
• C. $a+b$
• D. $b+c$

Answer 1

Answer: (C)

$a+b$

Given, $\angle C=\frac{\pi }{2}$
$\because \frac{c}{\sin C}=2R$ ....(By Sine rule)
$\Rightarrow \frac{c}{\sin \frac{\pi }{2}}=2R\Rightarrow c=2R$ ....(i)
and $\tan \frac{C}{2}=\frac{r}{s-c}\Rightarrow \tan \frac{\pi }{4}=\frac{r}{s-c}=1$
$\Rightarrow r=s-c\Rightarrow r=\frac{a+b+c}{2}-c$
$\Rightarrow 2r=a+b-c$
$\Rightarrow 2r=a+b-2R$ ....[Using equation (i)]
$\Rightarrow 2(r+R)=a+b$