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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
 #2
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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  laugh

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

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