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Common Internal Tangent

Date: 03/29/2002 at 14:12:54
From: Naveed Paya
Subject: Geometry

Question: Two circles with radii 9 and 6 are 2 cm. apart. Find the 
length of the common internal tangent.

I made the diagram but I can't figure out where to draw the right 
triangles to get the answer.

Thank you.

Date: 03/29/2002 at 16:59:38
From: Doctor Jubal
Subject: Re: Geometry

Hi Naveed,

Thanks for writing Dr. Math.

First, let me describe the diagram I'm going to use in explaining 
this. Points A and B are the centers of the two circles, and line 
segment AB connects them. Draw the common internal tangent and call 
its point of tangency with circle A point C, its point of tangency 
with circle B point D, and its intersection with segment AB point E.

The first thing we'll see is that triangle ACE is similar to triangle 
BDE. Because tangent lines are perpendicular to the circle's radius at 
the point of tangency, angle ACE is a right angle. Similarly, angle 
BDE is a right angle. Right angles are congruent to each other. Since 
segments AC and BD are both perpendicular to CD, they are parallel to 
each other. So, segment AB intesects the parallel segments AC and BD, 
making angle CAE congruent to angle DBE. Since triangles ACE and BDE 
have two pairs of congruent angles, they are similar triangles.

In similar triangles, corresponding parts are proportional to each 
other. Let's say that circle A is the bigger circle. Then AC = 9. The 
corresponding part of circle B is BD, which has length 6. Since AC is 
3/2 the size of BD and the two triangles are similar, any part of 
triangle ACE is 3/2 larger than the corresponding part of triangle 

We can use this to find the length of segment AE. We know that 
AB = AE+BE = 6+9+2 = 17. Also, we know that AE is 3/2 times the size 
of its corresponding part in the other triangle, BE. So we have this 
system of two equations:

  AE + BE = 17
  AE = (3/2)BE

Can you find values of AE and BE that satisfy these equations? If you 
can, then you can use the Pythagorean theorem to find the lengths of 
segments CE and DE, and their sum is the length of CD, the common 
internal tangent.

Does this help? Write back if you'd like to talk about this some more, 
or if you have any other questions.

- Doctor Jubal, The Math Forum   
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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