Question

In Figure 2. a right triangle ABC. circumscribes a circle of radius r. If AB and BC are of length 8 cm and 6 cm respectively. find the value of r.

Answer 1

Construction : Join OA, OB, OC, and draw perpendicular to AB and BC at P and Q respectively.


In right triangle ABC
By using Pythagoras theorem we have
AC² = AB² + BC²
= 8² + 6²
= 64 + 36
= 100
∴ AC² = 100
⇒ AC = 10 cm
Now,

$$\begin{gathered} \mathrm{Ar}\left(△\mathrm{ABC}\right)=\mathrm{Ar}\left(△\mathrm{AOB}\right)+\mathrm{Ar}\left(△\mathrm{BOC}\right)+\mathrm{Ar}\left(△\mathrm{AOC}\right) \\ \Rightarrow \frac{1}{2}\times \mathrm{AB}\times \mathrm{BC}=\frac{1}{2}\times \mathrm{OP}\times \mathrm{AB}+\frac{1}{2}\times \mathrm{OQ}\times \mathrm{BC}+\frac{1}{2}\times \mathrm{OR}\times \mathrm{AC} \\ \Rightarrow 8\times 6=x\times 8+x\times 6+x\times 10 \\ \Rightarrow x=2\mathrm{cm} \end{gathered}$$Ar(△ABC)=Ar(△AOB)+Ar(△BOC)+Ar(△AOC)⇒12×AB×BC=12×OP×AB+12×OQ×BC+12×OR×AC⇒5×12=x×5+x×12+x×13⇒60=30x⇒x=2 cm