Question

$ABC$ is a right angled triangle, right angled at $B$ such that $BC=6$ cm and $AB=8$ cm. A circle with centre $O$ is inscribed in $△ABC$. The radius of the circle is

• A. $1$ cm
• B. $2$ cm
• C. $3$ cm
• D. $4$ cm

Answer 1

The correct option is B $2$ cm

Let $ABC$ is the right angled triangle such that $\angle B=90$, $BC=6$ cm and $AB=8$ cm.

Let $O$ be the center and $r$ be the radius of the incircle.
$AB$, $BC$, and $AC$ are the tangents at $P$,$N$, $M$.
$\therefore OP=ON=OM=r$
Area of $△ABC=\frac{1}{2}\times 6\times 8$ $=24$
By Pythagoras Theorem, we have
$A{C}^2=A{B}^2+B{C}^2$
$\Rightarrow A{C}^2=8^2+6^2$
$\Rightarrow A{C}^2=64+36$
$\Rightarrow A{C}^2=100$
$\Rightarrow AC=10$
Area of $△ABC=$ Area $△OAB+$ Area $△OBC+$ Area $△OAC$
$\Rightarrow 24=\frac{1}{2}r\times AB+\frac{1}{2}r\times BC+\frac{1}{2}r\times AC$
$\Rightarrow 24=\frac{1}{2}r(AB+BC+AC)$
$\Rightarrow 48=r(8+6+10)$
$\Rightarrow r=\frac{48}{24}$
$\Rightarrow r=2$
Thus, radius of incircle is $2$ cm.