Answer:
1272.2 units²
Explanation:
This problem requires several steps.
1. The shaded area is the area of the circle minus the area of the triangle.
2. The area of the circle can be found using the radius. We are not given the radius, but using the right triangle, we can fond the diameter. Then the radius is half of the diameter.
3. The area of the triangle is base × height / 2. The base and the height are the given lengths of the legs.
Let's use the Pythagorean theorem with the triangle to find the diameter. The diameter is the hypotenuse of the right triangle.
a² + b² = c²
40² + 9² = c²
1600 + 81 = c²
c² = 1681
c = √1681
c = 41
The diameter of the circle is 41.
radius = diameter/2 = 41/2 = 21.5
The radius of the circle is 21.5
shaded area = area of circle - area of triangle
area of circle = πr²
area of triangle = bh/2
shaded area = πr² - bh/2
shaded area = (3.14159)(21.5)² - (40)(9)/2
shaded area = 1452.2 - 180
shaded area = 1272.2 units²