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In the figure, triangle $ABC$ is inscribed in a semicircle with diameter $AC$ of length $20$ inches, and $AB = 10$ inches. When the area of the shaded region, in square inches, is expressed in the form $a\pi - b,$ what is the value of $a + b\,?$

 

 Aug 22, 2023
 #1
avatar+128758 
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Area of semi-circle =  (1/2) pi (diameter / 2)^2  = (1/2) pi (10)^2  =  50pi      (1)

 

BC =   sqrt [ 20^2  -10^2 ] =  sqrt [ 300 ]  = 10sqrt 3 

 

Area of right triangle ABC  = (1/2) AB * BC =   (1/2) (10) (10sqrt 3)  = 50 sqrt 3       (2) 

 

Shaded area =   (1) - (2)   =    50 pi  - 50sqrt 3   

 

a + b =     50  + 50sqrt 3  ≈   136.6

 

 

cool cool cool

 Aug 22, 2023

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