To answer this question, we need:
1. Find the area of the sector with a 90-degree angle.
2. Find the area of the triangle in the figure.
3. Subtract the area of the 90-degree angles minus the area of the triangle.
We can proceed as follows:
Area of the sector with a 90-degree angleTo find it, we can use the next formula:
[tex]A_{\text{sector}}=\frac{n}{360}\cdot\pi\cdot r^2[/tex]Where, in this case, n = 90 (number of degrees in central angle of the sector). We also have that r = 16 units (the radius of the circle). Now, we can calculate this area:
[tex]A_{\text{sector}}=\frac{90}{360}\cdot\pi\cdot(16)^2=\frac{1}{4}\cdot\pi\cdot256=64\pi\approx201.0619\ldots[/tex]The area, in this case, is 64pi square units or 201.0619 square units.
Area of the triangle in the figureSince this is a right triangle, we have that the base = 16 units, and the height is equal to 16 too. Then, we have:
[tex]A_{\text{triangle}}=\frac{b\cdot h}{2}=\frac{16\cdot16}{2}=\frac{256}{2}\Rightarrow A_{triangle}=128[/tex]Now, we have that the area of the triangle is 128 square units.
Area of the Blue Region
This area is:
[tex]A_{\text{sector}}-A_{\text{triangle}}=201.0619-128=73.0619[/tex]Therefore, the area of the blue region is, approximately, equals to 73.0619 square units.
