Question

A $\Delta ABC$ is right angled at $B$. Then the diameter of the incircle of the triangle is

• A. $2(c+a-b)$
• B. $c+a-2b$
• C. $c+a-b$
• D. none of these

Answer 1

The correct option is C $c+a-b$

$2r=\frac{2\Delta }{s}=\frac{ac}{s}=\frac{2ac}{a+b+c}$

$=\frac{2ac(a+c-b)}{(a+c{)}^2-b^2}$
Now,

$\frac{1}{2r}=\frac{a^2+c^2-b^2}{2ac(a+c-b)}+\frac{2ac}{2ac(a+c-b)}$

$=cosC+\frac{1}{a+c-b}$

$=\frac{1}{a+c-b}(\because cosC=0)$

$\therefore 2r=a+c-b$