In triangle ABC, angle B = 90 degrees. Semicircles are constructed on sides AB, AC, and BC, as shown below. Show that the total area of the shaded region is equal to the area of triangle ABC.
To prove that the shaded area is equal to the area of a triangle (ABC), we have to find the area of the entire figure, and then subtract the area of the largest semicircle.
Let the sides of a triangle ABC be 3-4-5
Let denote the areas of 3 semicircles, starting with the smallest one: A1, A2, and A3
A1 = (1.52*pi) /2 = 3.534291735 u2
A2 = 2pi = 6.283185307 u2
A3 = (2.52*pi) /2 = 9.817477043 u2
ΔA = (3*4) /2 = 6 u2
ΔA = ( A1 + A2 + ΔA ) - A3
6 = 6