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In the diagram AB = 5, AC = 8 and BS = 25.  Find the area of the circle.

 

 Dec 8, 2020
 #2
avatar+117546 
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We can use the  secant-secant  product theorem  to find PC

 

AC (AC + PC)  = AB (AB + SB)

 

8 ( 8 + PC)  = 5 (5 + 25)

 

64 + 8PC  =  5 (30)

 

8PC =  150 - 64

 

8PC = 86

 

PC  =  86/8  =  43/4  =10.75

 

So   AP  = 8 + 10.75  = 18.75

 

So  SP^2  =   [ AS^2 - AP^2 ] =   [ 30^2  - 18.75^2] 

 

Connect   SC  and we  have right triangle  SCP....and since APS is a right angle....then SC will be the diameter of the  circle

 

So  SC   = sqrt  [ PC^2  + SP^2 ]  =  sqrt  [ 10.75^2  + 30^2  - 18.75^2  ]   = sqrt (664)

 

So....the radius =  sqrt (664) / 2

 

And the area of the  circle is     (sqrt (664) / 2 )^2 * pi  =   (664/4)  pi = 166 pi  units^2

 

 

cool cool cool

 Dec 8, 2020
 #3
avatar+1164 
+1

AC * AP = AB * AS

 

8(8 + PC) = 5 * 30

 

PC = 10.75

 

SP = sqrt(AS2 - AP2)

 

SC = sqrt(SP2 + PC2)

 

Area of a circle       A = (SC/2)2 * pi

 Dec 8, 2020

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