Question

In $△ABC$, $C=\frac{\pi }{2}$, show that 2(r + R) = a + b

Answer 1

$\frac{C}{\sin C}=2R$

$\Rightarrow C=90°$

$\Rightarrow C=2R..........\left(1\right)$

$\Rightarrow \tan \frac{C}{2}=\frac{r}{S-C}$

$\Rightarrow r=S-C$ $[\therefore \frac{C}{2}=45°,\tan \frac{C}{2}=1]$

$\Rightarrow r=\frac{a+b+c}{2}-C$

$\Rightarrow 2r=a+b-c.........\left(2\right)$

adding $\left(1\right)$ and $\left(2\right)$ we get

$\Rightarrow 2r+2R=a+b$

$2(r+R)=a+b$

Hence, proved.