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Externally tangent circles with centers at points $A$ and $B$ have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$, where $B$ is on $\overline{AC}$. What is the length $BC$?

 Mar 2, 2024
 #1
avatar+128794 
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Let  the tangent line touch circle A at D   and  circle B  at E

 

Let the distance from  B to C  = x

 

We can set up similar triangles ADC  and BEC  such that

 

AD / AC  =  BE / BC

 

5 / ( 5 + 3 + x)  =  3  / x

 

5 ( 8 + x)  = 3   / x          cross-multiply

 

5x  = 3 (8 + x)

 

5x  = 24 + 3x

 

2x  = 24

 

x = 12  = BC

 

 

cool cool cool

 Mar 3, 2024

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