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Two circles are externally tangent at T.  The line AB is a common external tangent to the two circles, and P is the foot of the altitude from T to line AB. Find the length AB.

 May 29, 2024
 #1
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First, we can graph this problem. 

 

We can use the Pythagorean Theorem to complete this problem. 

First, we need similar triangles. We can extend AB towards the direction of A and meet it up with the line connecting the radiuses. We can call this point O. 

Let the center of the unit circle be N and the center of the larger circle M. Let's let x be the distance from the left edge of the smaller circle to O. 

 

Notice that triangles BMO and ANO are similar triangles. This means we can write

BM/MO=AN/NO4/(4+2+x)=1/(1+x)4/(6+x)=1/(1+x)4(1+x)=1(6+x)4+4x=6+x3x=2x=2/3OA=NO2NA2=(1+2/3)212=25/91=16/9=4/3OB=MO2MB2=(6+2/3)242=400/916=256/9=16/3AB=OBOA=16/34/3=12/3=4

 

You can also use translation for this problem, but my small brain can't figure it out!

Thanks! :)

 May 30, 2024

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