ABC is a right triangle in which . If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle.

ABC is a right triangle in which $\angle B= 90^{\circ}$ . If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle.

Answers (1)

Given $\triangle ABC$ right angled at B
AB = 8 cm ; BC = 6cm

P $\rightarrow$ point of contact of circle and side AB
Q $\rightarrow$ point of contact of circle and BC
R $\rightarrow$ point of contact of circle and AC
O $\rightarrow$ centre of circle
Solution $\rightarrow$ $\triangle ABC$ is right angled $\triangle$
$\Rightarrow CA^{2}= AB^{2}+BC^{2}$
$\Rightarrow CA^{2}= 8^{2}+6^{2}$
$\Rightarrow CA= \sqrt{100}$
$\Rightarrow CA= 10$ cm
Area of $\triangle ABC= \frac{1}{2}\times AB\times BC= \frac{1}{2}\times 8\times 6= 24\, cm^{2}$
Area of $\triangle ABC= area\, of\, \left ( \triangle AOC+\triangle AOB+\triangle BOC \right )$
$= \frac{1}{2}\times OR\times AC+\frac{1}{2}\times OP\times AB+\frac{1}{2}\times OQ\times BC$
Also $OR= OP= OQ= r$ $\left ( r= radius \, of\, circle \right )$
$24\, cm^{2}= \frac{1}{2}\times r\times 10+\frac{1}{2}\times 4\times 8+\frac{1}{2}\times r\times 6$
$24= \frac{1}{2}\times 24r$
$\Rightarrow r= 2cm\rightarrow$ radius of circle