In the given figure, △ 𝐴𝐵C is in the semi circle. Find the area of the shaded region given that 𝐴𝐵 = 42 𝑐m

It is given that 𝐴𝐵 = 42𝑐𝑚 

𝐴𝐵 is the diameter of the semi-circle and it is also the base of the triangle 𝐴𝐵𝐶. 

𝑂𝐶 is the radius of the semi-circle and it is also the height of the triangle 𝐴𝐵𝐶. So,

$\begin{array}{r}\overline{OC}=\frac{1}{2}AB\\ =\frac{42}{2}\mathrm{cm}\\ =21\mathrm{cm}\end{array}$

Now, area of the shaded region =area of the semi-circle − area of the triangle area of semi-circle= $\frac{{\mathrm{\pi r}}^2}{2}$

$\begin{array}{r}=\frac{22}{7}\times 21\times 21\times \frac{1}{2}\\ =11\times 3\times 21\\ =693{\mathrm{cm}}^2\end{array}$

Now, area of the triangle = $\frac{1}{2}\times \text{ base }\times \text{ height }$

$\begin{array}{r}=\frac{1}{2}\times AB\times OC\\ =\frac{1}{2}\times 42\times 21\\ =441{\mathrm{cm}}^2\end{array}$

So, the area of the shaded region = (693 − 441) 𝑐𝑚² 

= 252𝑐𝑚²