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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.

 Jun 8, 2020
 #1
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The circle with center A has a radius of 10.

The circle with center B has a radius of 7.

These two circles are externally tangent; therefore, AB = 17.

 

There is an external tangent to these two circles. Both circles lie on the same side of this external tangent.

Label this external tangent XYA, where X is the point of tangency to circle A and Y is the point of tangency 

to circle B. A is the point where this line intersects line AB.

 

Let x represent the length BC.

 

Triangle(AXC) is similar to triangle(BYC) by AA.

We have:  AX / AC  =  BY / BC     --->     10 / (17 + x)  =  7 / x

              cross-multiply:                                         10x  =  7(17 + x)

                                                                              10x  =  119 + 7x

                                                                                3x  =  119

                                                                                  x  =  119/3

 Jun 8, 2020

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