Question

In an isosceles right angled triangle $△ABC$, $r=$ _________.

• A. $2$
• B. $1$
• C. $1-\frac{1}{\sqrt{2}}$
• D. $2-\sqrt{2}$

Answer 1

The correct option is C $1-\frac{1}{\sqrt{2}}$

$△ABC$ is isosceles right angle triangle.
$\therefore AB=BC=1$
$Now,A{C}^2=A{B}^2+B{C}^2$
$=1+1=2$
$\therefore AC=\sqrt{2}$
$\therefore$ Inradius of $△ABC$
$r=\frac{AB+BC-AC}{2}$
$=\frac{1+1-\sqrt{2}}{2}=\frac{2-\sqrt{2}}{2}$
$=1-\frac{\sqrt{2}}{2}=1-\frac{1}{\sqrt{2}}$