Tangent Line and CirclesDate: 04/05/99 at 12:33:37 From: Tom Hammond Subject: Length of segment of line tangent to two tangent circles of different radii Two circles of different radius are tangent to each other. A line is drawn that is tangent to both circles. What is the length of the segment between the two points of tangency of the line and the circles? My sophomore son and I have been trying to answer this. I have usually gotten through these pretty easily, but this one has us totally stumped. Thank you so much for your help! Tom Date: 04/05/99 at 17:03:50 From: Doctor Peterson Subject: Re: Length of segment of line tangent to two tangent circles of different radii Hi, Tom. This is a nice little problem that involves some useful ideas. Draw the two circles and their tangent, and the line joining their centers. Then draw a radius to each point of tangency, forming a trapezoid ABCD. The angles at C and D are right, so AD and BC are parallel. Draw BE parallel to the tangent, and think about the triangle ABE. You should be able to find a relation that will give you the length CD from the radii AD and BC. \ \ D ***********\ *** /*** \ ** / ** \ C * E+ * *****\ * / \ * ** /** \ * / **\ / * ----*------------+------------*-----+-----*---- * A ** B * * * ** ** * * ***** ** ** *** *** *********** Let me know if you need more help. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/05/99 at 21:06:44 From: thammond Subject: Re: Length of segment of line tangent to two tangent circles of different radii Thanks for the help, but I've run up against a brick wall. The right triangle ABE has a hypotenuse of the sum or the radii (R1+R2) and a shorter leg of the difference between the radii (R2-R1), but the other two angles are unknown and the length of the third side (same as the length of the segment between the points of tangency) cannot be determined... ouch! Help? Thanks! Tom R2-R1 ____________ | / | / | / | / R2+R1 | / | / | / |/ Date: 04/05/99 at 22:38:03 From: thammond Subject: Re: Length of segment of line tangent to two tangent circles of different radii I feel like a total idiot... and Pythagoras would be totally ashamed of me. We made this problem up ourselves and then couldn't answer it, but... well, I sent a previous note saying your help didn't complete the process, but we had to remember that the segment of the tangent line could only be put in terms of the radii. So... The hypotenuse is the sum of the radii (R1+R2). The one leg of the right triangle created by your method is the difference between the radii (R2-R1). If we set the other leg, that is, the one congruent to the segment we are trying to find, equal to "x", then Pythagoras shows us that X equals 2 times the square root of the product of the radii. X^2 + (R2-R1)^2 = (R1+R2)^2 X = 2*(SQRT)(R1R2) You perform a wonderful and needed service. Thank you SO much! Tom Hammond Date: 04/06/99 at 12:00:08 From: Doctor Peterson Subject: Re: Length of segment of line tangent to two tangent circles of different radii Good work, Tom! And you thought this problem up yourselves? Most problems I think up myself can't be solved! This is actually a fairly standard problem, in part because of the neat answer, as well as the neat method. Yes, Pythagoras is the magic word in a surprising number of problems. Did you also notice that the answer is the geometric mean of the diameters of the circles? It's surprising how often this turns up in geometry, explaining the name "geometric." - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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