The area of the shaded region is 46.17 ft.²
The given parameters are;
The radius of the semicircle, r = 9 ft.
The height of the triangle in the semicircle, h = The radius of the semicircle = 9 ft.
The location of the triangle = Inscribed in the semicircle
The base length of the triangle, b = The diameter of the semicircle
Required parameter = The area of the shaded region
Method;
Find and subtract the area of the triangle from the area of the semicircle
The diameter of the semicircle = 2 × The radius of the semicircle
∴ The diameter of the semicircle = 2 × 9 ft. = 18 ft.
The base length of the triangle, b = The diameter of the semicircle = 18 ft.
The area of the triangle, A = (1/2)·b·h
∴ A = (1/2) × 18 ft. × 9 ft. = 81 ft.²
The area of the semicircle, [tex]\mathbf{A_{sc}}[/tex] = Area of circle/2 = π·r²/2
[tex]\mathbf{A_{sc}}[/tex] = π·r²/2
Where;
π = 3.14
∴ [tex]\mathbf{A_{sc}}[/tex] = 3.14 · (9 ft.)²/2 = 127.17 ft.²
The area of the shaded region, [tex]\mathbf{A_{r}}[/tex] = [tex]\mathbf{A_{sc}}[/tex] - A
∴ [tex]\mathbf{A_{r}}[/tex] = 127.17 ft.² - 81 ft.² = 46.17 ft.²
The area of the shaded region, [tex]A_{r}[/tex] = 46.17 ft.²
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