Question

In an isosceles right angled triangle $\Delta ABC$, r = ..................

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

Answer 1

Answer: (C)

$1-\frac{1}{\sqrt{2}}$

In $\Delta ABC,m\angle B={90}^o,AB=1$ and $BC=1$
$\therefore A{C}^2=A{B}^2+B{C}^2$ (Pythagoras theorem)
$=1^2+1^2$
$=1+1$
$=2$
$\Rightarrow AC=\sqrt{2}$
Now, radius r of a circle touching all the three sides of triangle
$=\frac{AB+BC-AC}{2}$
$=\frac{2-\sqrt{2}}{2}$
$=1-\frac{1}{\sqrt{2}}$