Question

The given figure shows a circle (with centre $O$ ) inscribed in a right-angled triangle $ABC$ with $\angle B={90}^{\circ },AB=8cm$ and $BC=6cm$. Find:
i) the radius of the circle
ii) the area of the shaded portion enclosed between the circle and the triangle.

Answer 1

$\begin{array}{c}\left(1\right)\\ AC=\sqrt{6^2+8^2}=10cm\\ s=\frac{10+6+8}{2}=12\\ r=\sqrt{\frac{\left(5-a\right)\left(5-b\right)\left(5-c\right)}{s}}\\ =\sqrt{\frac{2\times 6\times 4}{12}}\\ =2\left(radiusofinscribedcircle\right)\\ \left(2\right)\\ areaofshadedportion=areaof\Delta -areaofcircle\\ =\frac{1}{2}\times 8\times 6-\pi {\left(2\right)}^2\\ =24-4\pi \\ =4\left(6-\pi \right)c{m}^2\end{array}$