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 BDE. 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 http://mathforum.org/dr.math/
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