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The circle with center $A$ and radius $10$ and the circle with center $B$ and radius $7$ are externally tangent. A line that is externally tangent to both circles is drawn, where both circles lie on the same side of the line. This common tangent intersects line $AB$ at $C$ Find the length $BC$

 Apr 17, 2021
 #1
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BC = 79/2.

 Apr 17, 2021
 #2
avatar+118586 
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See the following image

 

 

 

 

Let  FC  = x

BC =  7+ x

BE  =  7

AC = 24  + x

AD = 10

 

So....by similar  triangles  CBE  and CAD

 

CB/ BE  = CA/ AD

 

(7 + x)/ 7  =  (24 + x) / 10

 

10 (7 + x)  = 7(24 + x)

 

70  +  10x  =  168  + 7x

 

3x  =  168  -  70

 

x  =  98/3  = FC

 

So

 

BC  =  7  + 98/3   =   119/3

 

cool cool cool

 Apr 17, 2021

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