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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 TP. 

 Jun 23, 2024
 #2
avatar+129725 
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Let O be the center of the larger circle and let N be the center of the smaller circle

 

Extend   BA   and ON  to meet at  M

 

Let MN = 3 + x      MT = 6 + x      OM =  13 + x    NA =  3    BO = 7

 

First relationship

 

MN / NA  = MT / TP

 

NA / MN  = TP  / MT

 

3 / (3 + x)  = TP /  (6 + x)

 

TP  = 3(6 + x) / (3 + x)

 

Second relationship

 

TP / MT =  BO / OM

 

3 (6 + x) /  (3 + x)         7

______________  = ________

        (6 + x)               (13 + x)

 

 

3 / ( 3 + x)  = 7 / (13 + x)

 

3(13 + x) = 7(3 + x)

 

39 + 3x  = 21 + 7x

 

18  = 4x

 

x = 18/4  = 9/2

 

TP  = 3(6 + 9/2) / (3 + 9/2)  =  21 / 5 =  4.2

 

CORRECTED  !!!!

 

cool cool cool

 Jun 24, 2024
edited by CPhill  Jun 24, 2024
edited by CPhill  Jun 24, 2024
 #3
avatar+33 
+1

You had a typo in 39+3x=31+7x it should be 21 instead of 31. so x=9/2 so TP=21/5. Thank you for all your help

jumbean  Jun 24, 2024
edited by jumbean  Jun 24, 2024

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