Answer:
[tex]A \approx 1140.3\ units^2[/tex]
Step-by-step explanation:
To solve this problem we first need to find the diameter of the circle. We are told the point [tex]N[/tex] is the circle, so we can assume it is the center of the circle. To find the diameter we can use the Pythagorean Theorem:
[tex]a^2 + b^2 = c^2\\40^2 + 9^2 = c^2\\1600 + 81 = c^2\\1681 = c^2\\\sqrt{1681} = c\\41 = c[/tex]
The diameter is [tex]41[/tex] units, which means the radius is [tex]\frac{41}{2}[/tex]. From here we can calculate the area of the circle:
[tex]A_c = \pi r^2\\A_c = \pi (\frac{41}{2})^2\\A_c = \pi 420.25\\A_c \approx 1320.25[/tex]
Now we need to subtract the area of the triangle from the area of the circle. Calculate the area of the triangle:
[tex]A_t = \frac{bh}{2}\\A_t = \frac{40 \times 9}{2}\\A_t = \frac{360}{2}\\A_t = 180[/tex]
Subtract the triangle area from the circle area to get the area of the shaded area:
[tex]A = A_c - A_t\\A = \pi420.25 - 180\\A = 1140.25431\\A \approx 1140.3\ units^2[/tex]
