Question

ABC is a right angled triangle with AB = 12 cm and AC = 13 cm. A circle, with centre O, has been inscribed inside the triangle.

Calculate the value of x, the radius of the inscribed circle.

• A. 8 cm
• B. 3 cm
• C. 2 cm
• D. 9 cm

Answer 1

The correct option is C

2 cm

In $\Delta ABC,\angle B={90}^{\circ }$

$OL\perp AB,OM\perp BC$ and
$ON\perp AC.$

$\therefore$ LBMO is a square.

LB = BM = OL = OM = ON = $x$

$\therefore AL=12-x$

$\therefore AL=AN=12-x$ ( Tangents from an external point)

$\because \Delta ABC$ is a right triangle

$\therefore A{C}^2=A{B}^2+B{C}^2$

$\Rightarrow (13{)}^2=(12{)}^2+B{C}^2$

$\Rightarrow 169=144+B{C}^2$
$\Rightarrow B{C}^2=169-144=25$

$\therefore BC=\sqrt{25}=5cm$

$MC=BC-x$

$\therefore MC=5-x$

But CM = CN

$\therefore CN=5-x$

Now AC = AN +NC

$\Rightarrow 13=12-x+5-x\Rightarrow 13=17-2x$

$2x=17-13=4$

$\therefore x=\frac{4}{2}=2cm$