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avatar+1347 

Plz help with semicircles

 

Semicircles are drawn on diameters $\overline{AB}$ and $\overline{CD},$ as shown below.   Find the area of the shaded region.

 

 Aug 18, 2023
 #2
avatar+128732 
+1

 

 

 

 

 

EF  =    sqrt (2^2 - (sqrt 3)^2) =  sqrt ( 4 -3) =  sqrt (1) =  1

 

Area of triangle  FBE = (1/2) (AB) (EF) = (1/2)(2sqrt 3) ( 1) =  sqrt (3)

 

tan (angle  BFE)  =  sqrt (3) / 1  = sqrt (3)

 

arctan (sqrt (3)) = 60° = measure of angle BFE

 

So  angle AEB = 120°

 

Area of sector AEB =  pi * 2^2 / 3  = (4/3)pi

 

Area between sector AEB and triangle FBE =  (4/3)pi - sqrt (3)         (1)

 

Area  of small semi-circle  =   pi (sqrt 3)^2 / 2 =  (3/2) pi                     (2)

 

Shaded area =  (2) - (1)  =   (3/2) pi - [ (4/3) pi - sqrt (3) ]  =   pi/6 + sqrt (3)

 

 

cool cool cool

 Aug 19, 2023

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