what is the area of the shaded portion of the circle? (16π - 32) in² (16π - 8) in² (64π - 32) in² (64π - 8)…

Explanation:

Step1: Calculate the area of the sector

The central - angle of the sector is 90° (since it's a right - angled sector). The formula for the area of a sector of a circle is $A_{sector}=\frac{\theta}{360}\times\pi r^{2}$, where $\theta$ is the central angle and $r$ is the radius. Given $r = 8$ in and $\theta=90$, then $A_{sector}=\frac{90}{360}\times\pi\times8^{2}=\frac{1}{4}\times\pi\times64 = 16\pi$ in².

Step2: Calculate the area of the triangle

The triangle is a right - angled triangle with base and height equal to the radius of the circle. The formula for the area of a triangle is $A_{triangle}=\frac{1}{2}\times base\times height$. Here, base = height = 8 in, so $A_{triangle}=\frac{1}{2}\times8\times8=32$ in².

Step3: Calculate the area of the shaded region

The area of the shaded region is the area of the sector minus the area of the triangle. So $A = A_{sector}-A_{triangle}=16\pi - 32$ in².

Answer:

$(16\pi - 32)$ in²