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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!

Jul 17, 2020

#1
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Here's a hint: draw the other half of the semicircle, and drop perpendiculars.

Jul 18, 2020
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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

Jul 18, 2020
edited by Dragan  Jul 18, 2020
edited by Dragan  Jul 18, 2020