Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively. If $r$ is the inradius of the triangle

Then $r=?$

• A. $(s-a)\tan \frac{B}{2}$
• B. $(s-b)\tan \frac{B}{2}$
• C. $(s-b)\tan \frac{C}{2}$
• D. $(s-a)\tan \frac{C}{2}$

Answer 1

The correct option is B $(s-b)\tan \frac{B}{2}$

$r=\frac{\Delta }{s}=\frac{\sqrt{s(s-a)(s-b)(s-c)}}{s}$

$=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}=(s-b)\sqrt{\frac{(s-a)(s-c)}{s(s-b)}}$

We know that $\tan \frac{B}{2}=\frac{(s-c)(s-a)}{s(s-b)}$

Thus, $r=(s-b)\tan \frac{B}{2}$