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In triangle ABC, B = 90 degrees. Semicircles are constructed on sides AB, AC, and BC, as shown below. Prove that the total area of the shaded region is equal to the area of triangle ABC.

 

 Apr 6, 2020
edited by Guest  Apr 6, 2020
 #1
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Note  that  the area of the  right  triangle = (1/2)(BC) (AB)

 

And note  that the  area  between  the right triangle and  the  circle  =  (1/2)pi (AC/2) ^2  -  (1/2) (BC)(AB)  =

 

(1/2) [ pi  ( AC)^2  / 4  - (BC)(AB)  ]  =    (1/2) [ BC^2 + AB^2 ] / 4  -  (1/2) (BC)(AB)          (2)

 

The area  of  the two  semi-circles  =  (1/2)  [  (BC/2)^2  +  (AB/2)^2  ]  =  (1/2)  [ BC^2 + AB^2 ]/ 4   (3)

 

So....the shaded area  =  (3)  - (2)  =

 

(1/2) [ BC^2 + AB^2   ] /4  -   [ (1/2) (BC^2  + AB^2 ]   / 4    - (1/2) (BC)(AB)  ]  =

 

(1/2) (BC)(AB)    which  is  the  same  as  the area of  the right triangle

 

 

cool cool cool

 Apr 6, 2020
 #2
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THx so much!!!!

Guest Apr 6, 2020

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