Answer: The correct option is (A) [tex](16\pi-32)~\textup{in}^2.[/tex]
Step-by-step explanation: We are given to select the area of the shaded portion of the circle shown in the figure.
From the figure, wee that
there is a circle of radius, r = 8 in.
So, the area of the whole circle will be
[tex]A_c=\pi r^2=\pi\times 8^2=64\pi~\textup{in.}^2[/tex]
Now, the triangle shown is a right-angled one with base length 8 in and height 8 in.
So, the area of the triangle is given by
[tex]A_t=\dfrac{1}{2}\times 8\times8=32~\textup{in.}^2[/tex]
Since the area of the shaded portion is equal to one fourth of the area of circle minus the area of the triangle, so we get
[tex]Area~of~the~shaded~portion\\\\=\dfrac{1}{4}\times A_c-A_t\\\\=\dfrac{1}{4}\times 64\pi-32\\\\=(16\pi-32)~\textup{in}^2.[/tex]
Thus, the required area of the shaded portion is [tex](16\pi-32)~\textup{in}^2.[/tex]
Option (A) is CORRECT.
