To find the area of the shaded region, we need to find the area of triangle ABC and subtract the area of sector AEC.
First, we can use the Pythagorean theorem to find the length of side BC:
BC^2 = AB^2 - AC^2
BC^2 = 15^2 - 8^2
BC^2 = 169
BC = 13
Now we can find the area of triangle ABC using the formula:
area = (1/2) * base * height
area = (1/2) * 15 * 8
area = 60
To find the area of sector AEC, we need to find the measure of angle AEC. Since triangle ABC is inscribed in circle E, we know that angle AEC is a central angle that intercepts arc AC. The measure of angle AEC is therefore equal to half the measure of arc AC.
The circumference of circle E is 2πr, where r is the radius. Since triangle ABC is inscribed in circle E, the diameter of circle E is equal to side AC. The radius of circle E is therefore half the length of AC:
r = (1/2) * AC
r = (1/2) * 15
r = 7.5
The circumference of circle E is 2πr:
circumference = 2πr
circumference = 2π(7.5)
circumference = 15π
Since arc AC is one-third of the circumference of circle E, its measure is:
arc AC = (1/3) * circumference
arc AC = (1/3) * 15π
arc AC = 5π
The measure of angle AEC is therefore:
angle AEC = (1/2) * arc AC
angle AEC = (1/2) * 5π
angle AEC = (5/2)π
To find the area of sector AEC, we can use the formula:
area = (1/2) * r^2 * θ
area = (1/2) * 7.5^2 * (5/2)π
area = (1/2) * 56.25 * 2.5π
area = 70.3125π
Finally, we can find the area of the shaded region by subtracting the area of sector