+185 In triangle \(ABC, \angle B = 90^\circ.\) Semicircles are constructed on sides \(\overline{AB}, \overline {AC}\) and \(\overline{BC}\) as shown below. Show that the total area of the shaded region is equal to the area of triangle \(ABC.\)
Please explain clearly if you can, thank you!
+1486 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 
