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The figure shows circle O with diameter AB and inscribed angle ABC with AB = 8 and BC = 4. When the area of the shaded region, in square units, is expressed in the form a + bπ, what is the value of ab?

 

 Jul 20, 2022
 #1
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In triangle ABC, angle  BCA  = 90° 

 

And the shaded area is composed of equilateral triangle BOC (BO = BC = CO  = 4)   plus sector COA

 

Area of triangle   BOC  = (1/2) 4^2  sin 60°  =  8sqrt (3) / 2  =    4sqrt (3)

 

Area of sector COA  = (1/3)area of circle =   (1/3) pi (4^2)  = (16/3) pi 

 

Total area =   4sqrt (3)  +  (16/3) pi

 

ab =     (4sqrt 3) ( 16/3)  =  64 / sqrt 3

 

 

cool cool cool

 Jul 20, 2022

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