Question

In a right triangle ABC, right angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle is [CBSE 2014]
(a) 1 cm
(b) 2 cm
(c) 3 cm
(d) 4 cm

Answer 1

In right triangle ABC
By using Pythagoras theorem we have
AC² = AB² + BC²
= 5² + 12²
= 25 + 144
= 169
∴ AC² = 169
⇒ AC = 13 cm
Now,

$$\begin{gathered} \mathrm{Ar}\left(△\mathrm{ABC}\right)=\mathrm{Ar}\left(△\mathrm{AOB}\right)+\mathrm{Ar}\left(△\mathrm{BOC}\right)+\mathrm{Ar}\left(△\mathrm{AOC}\right) \\ \Rightarrow \frac{1}{2}\times \mathrm{AB}\times \mathrm{BC}=\frac{1}{2}\times \mathrm{OP}\times \mathrm{AB}+\frac{1}{2}\times \mathrm{OQ}\times \mathrm{BC}+\frac{1}{2}\times \mathrm{OR}\times \mathrm{AC} \\ \Rightarrow 5\times 12=x\times 5+x\times 12+x\times 13 \\ \Rightarrow 60=30x \\ \Rightarrow x=2\mathrm{cm} \end{gathered}$$
Hence, the correct answer is option (b).