Question

$ABC$ is a right angle triangle right angled at $A.$ A circle is inscribed in it. The lengths of two sides containing the right angle are $6$ cm and $8$ cm. Find the radius of the circle$?$

• A. $3$ cm
• B. $2$ cm
• C. $5$ cm
• D. $4$ cm

Answer 1

The correct option is B $2$ cm

Let $△ABC$ be right angled at $A.$
Since the incentre is equidistant from the sides, let the radius of the incircle be $r.$
$\therefore$ $OP=OQ=OR=r$ cm
By Pythagoras' Theorem
$A{C}^2+A{B}^2=B{C}^2$
$\Rightarrow$ $B{C}^2=6^2+8^2=36+64=100$
$\Rightarrow$ $BC=10$ cm
Area of $△ABC=$Area of $△OAB+$Area of $△OBC+$Area of $△OCA$
$\Rightarrow$ $\frac{1}{2}\times AB\times AC=\frac{1}{2}\times r\times BC+\frac{1}{2}\times r\times BC+\frac{1}{2}\times r\times CA$
$\Rightarrow$ $\frac{1}{2}\times 6\times 8=\frac{1}{2}\times r\times 6+\frac{1}{2}\times r\times 10+\frac{1}{2}\times r\times 8$
$\Rightarrow$ $12r=24$

$\Rightarrow r=2$ cm.