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!

Otterstar Jul 17, 2020

#2**+2 **

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: A_{1}, A_{2}, and A_{3}

A_{1} = (1.5^{2}*pi) /2 = 3.534291735 u^{2}

A_{2} = 2pi = 6.283185307 u^{2}

A_{3} = (2.5^{2}*pi) /2 = 9.817477043 u^{2}

ΔA = (3*4) /2 = 6 u^{2}

**ΔA = ( A _{1} + A_{2} + ΔA ) - A_{3}**

**6 = 6 **

Dragan Jul 18, 2020