First, we find the area of the circle.
[tex]A=\pi r^2[/tex]
However, first, we have to find the diameter of the circle, which is the hypothenuse of the right triangle.
[tex]\begin{gathered} h^2=21^2+20^2 \\ h=\sqrt[]{441+400} \\ h=\sqrt[]{841} \\ h=29 \end{gathered}[/tex]
So, the diameter is 29 inches long. Remember that the radius is half the diameter, 14.5 inches long. Let's find the area of the circle.
[tex]A=\pi(14.5)^2=660.52in^2[/tex]
Now, we find the area of the triangle.
[tex]A=\frac{bh}{2}=\frac{20\cdot21}{2}=210in^2[/tex]
At last, we subtract the area of the circle and the area of the triangle to find the shaded region.
[tex]A_{\text{shaded}}=660.52-210=450.52in^2[/tex]
Hence, the answer is the first option.
