Question

ABC is a right-angled triangle whose $\angle A={90}^{\circ },AB=12cmandAC=16cm.$
The radius of incircle of $\Delta ABC$ is

• A.
  4 cm
• B.
  8 cm
• C.
  12 cm
• D.
  16 cm

Answer 1

The correct option is A

4 cm
Given that ABC is a right triangle with $\angle A={90}^{\circ },AB=12cmandAC=16cm.$



Let the radius of the incircle of $\Delta ABC$ be r.

In $\Delta ABC$, by Pythagoras theorem,

$B{C}^2=A{B}^2+A{C}^2\Rightarrow B{C}^2=(12{)}^2+(16{)}^2\Rightarrow B{C}^2=144+256\Rightarrow B{C}^2=400\Rightarrow BC=20cm$

Now, $areaof\Delta ABC=\frac{1}{2}\times AB\times AC=\frac{1}{2}\times 12\times 16=96c{m}^2$

Also, area of $\Delta ABC$ = area of $\Delta OAB$ + area of $\Delta OCB$ + area of $\Delta OCA$

$\Rightarrow 96=\frac{1}{2}\times OP\times AB+\frac{1}{2}\times OQ\times BC+\frac{1}{2}\times OR\times AC\Rightarrow 96=\frac{1}{2}\times r\times 12+\frac{1}{2}\times r\times 20\times r\times 16(\because OP=OQ=OR=r)\Rightarrow 96=24r\Rightarrow r=4cm$


Thus, the radius of the incircle of $\Delta ABC$ is 4 cm.

Hence, the correct answer is option a.