Question

In the given figure ∆ABC is right-angled at A, with AB = 6 cm and AC = 8 cm. A circle with centre O has been inscribed inside the triangle. Find the value of r, the radius of the inscribed circle.

Answer 1

Join OC, OA and OB. This gives triangles OAC, OAB and OCB.

Consider $∆$CAB. Using Pythagoras' theorem, we have:

$$\begin{gathered} {\left(CA\right)}^2+{\left(AB\right)}^2={\left(BC\right)}^2 \\ \Rightarrow 8^2+6^2=64+36=100 \\ \Rightarrow BC=10\mathrm{cm} \end{gathered}$$

Also,
$$\begin{gathered} \mathrm{Ar}\left(∆ABC\right)=\frac{1}{2}\times 6\times 8 \\ =24{\mathrm{cm}}^2 \end{gathered}$$

Further, we have:

$$\begin{gathered} \mathrm{Ar}\left(∆ABC\right)=\mathrm{Ar}\left(∆OAB\right)+\mathrm{Ar}\left(∆OCB\right)+\mathrm{Ar}\left(∆OCA\right) \\ \Rightarrow 24=\frac{1}{2}r\times 6+\frac{1}{2}r\times 8+\frac{1}{2}r\times 10 \\ \Rightarrow 24=\frac{1}{2}r\left(6+8+10\right) \\ \Rightarrow 24=\frac{1}{2}\times r\times 24 \\ \Rightarrow 24=12r \\ \Rightarrow 12r=24 \\ \Rightarrow r=2\mathrm{cm} \end{gathered}$$