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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.

 

 Mar 26, 2024
 #1
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Set the center of the small circle to be O1, and the center of the larger circle to be O2.

Draw a perpendicular line passing through O1, perpendicular to BO2, label the foot of the perpendicular point C.

ABCO1 is a rectangle, and so AB = CO1.

Also, CO2 = 4 - 1 = 3

Apply the pythagorean theorem, so (CO1)2=(O1O2)2-(CO2)2=25 - 9=16.

Therefore, CO1=AB=4.

 Mar 26, 2024

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