+0

# Geometry

+1
1
3
+33

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
+129725
+1

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  !!!!

Jun 24, 2024
edited by CPhill  Jun 24, 2024
edited by CPhill  Jun 24, 2024
#3
+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