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For the triangle below, let $x$ be the area of the circumcircle, and let $y$ be the area of the incircle.  Compute $x - y.$

 

 Jan 9, 2024
 #1
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c/ sin (105)  = 7/ sin (35)

 

c  =  7sin (105) / sin (35)  ≈  11.8

 

a / sin ( 40 )  = 7/sin (35)

 

a  =  7 sin (40) / sin (35) ≈ 7.8 

 

Semi-perimeter  of triangle  = (7 + 11.8 + 7.8) /  2  = 13.3 = S

 

Area of triangle  sqrt [ 13.3 (13.3- 11.8)(13.3 - 7.8) (13.3 - 7) ]   ≈  26.3 =  A

 

Incircle radius  =     A  /  S  =   26.3 / 13.3   ≈ 1.98

 

Circumcircle  radius  =   abc   /  ( 4A) =   (7 * 7.8 * 11.8)  /  [4 * 26.3 ]  ≈  6.12

 

x - y  =   pi ( 6.12^2 - 1.98^2)   ≈  105.35  

 

 

cool cool cool

 Jan 9, 2024
edited by CPhill  Jan 9, 2024

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