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June 25th, 2011, 10:53 PM  #11 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 
Let be the length from the vertex of the smaller base to the tangent point of the smaller circle. Draw a line that passes through the centres of two circles. Now shift it units until the end of the line touches the vertex of smaller base. Use Pythagorean theorem to determine the height Since the line is shifted from where it crosses the two centres, height equals to the sum of two diameters. Hence the height Smaller base is Hence to determine the larger base use the fact that larger base:smaller base = 9:1 by using the tangent line of externally tangent circles as midbase, two trapezoids are similar by 1:3. Smaller base:midbase = 1:3 and midbase:larger base = 1:3 = 3:9 therefore smaller base:larger base = 1:9. To determine the larger base, multiply the smaller base by 9 to get 
June 26th, 2011, 08:00 AM  #12 
Newbie Joined: Jun 2011 Posts: 21 Thanks: 0  Re: Finding Sides and area of trapezoid around two circles.
I thought the height was just 8?

June 26th, 2011, 12:09 PM  #13 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Finding Sides and area of trapezoid around two circles.
My inclination was to approach this problem with analytic geometry (and like johnny assume an isosceles trapezoid). I oriented the system such that the smallest side of the trapezoid lies on the yaxis and the side opposite lies on the line x = 8. The centers of the circles lie on the xaxis where the smaller circle is then given by (x  1)² + y² = 1 and the larger circle is given by (x  5)² + y² = 3². Now, the top side of the trapezoid must be tangent to the two circles. Let be the point of tangency with the smaller circle and be the point of tangency with the larger circle. We have that the line segments: and are parallel and both perpendicular to the top side of the trapezoid. Thus, we have: Square through: Recall and giving: Take the positive root: Now, substitute for : Thus, the slope of the top side of the trapezoid is: Recall this slope is perpendicular to the segment giving: Recall though that: Equating, we find: Combine like terms: Thus the slope m of the top side is: Now, using the pointslope formula, we find the equation of the line representing the top side as: Put into slopeintercept form, we have: Thus: The smaller base b is The larger base B is =\frac{18}{\sqrt{3}}=6\sqrt{3}" /> The two sides s are The area A is Note that these results agree completely with the results given by johnny, although I did not bother to rationalize the denominators. 
June 26th, 2011, 12:18 PM  #14  
Senior Member Joined: Feb 2010 Posts: 707 Thanks: 142  Re: Finding Sides and area of trapezoid around two circles. Quote:
 
June 29th, 2011, 10:52 PM  #15 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Finding Sides and area of trapezoid around two circles.
I thought it might be fun (at least I thought so at first until I spent a great deal of time chasing down silly mistakes! ) to generalize a bit, and let the smaller circle have radius r and the larger circle have radius kr where k ? 1. As before, we will assume an isosceles trapezoid and orient the system such that the smallest side of the trapezoid lies on the yaxis and the side opposite lies on the line x = 2r(k + 1). The centers of the circles lie on the xaxis where the smaller circle is then given by (x  r)² + y² = r² and the larger circle is given by (x  r(2 + k))² + y² = (kr)². Now, the top side of the trapezoid must be tangent to the two circles. Let be the point of tangency with the smaller circle and be the point of tangency with the larger circle. We have that the line segments: and are parallel and both perpendicular to the top side of the trapezoid. Thus, we have: Square through: Recall and giving: Take the positive root: Now, substitute for : Thus, the slope of the top side of the trapezoid is: Recall this slope is perpendicular to the segment giving: Crossmultiplication yields: Recall that so we have: Expansion, simplification, and solving for yields: Thus, the slope m of the top side is: Now, using the pointslope formula, we find the equation of the line representing the top side as: Put into slopeintercept form, we have: Thus: The smaller base b is The larger base B is The two sides s are The area A is 
June 30th, 2013, 10:09 AM  #16 
Newbie Joined: Jun 2013 Posts: 2 Thanks: 0  Re: Finding Sides and area of trapezoid around two circles.
Ive seen people say the smaller base is sqrt(3) / 3 by using the pythagorean theorem. Can someone go into a little more depth and explain how to use the pythagorean theorem on this problem when you only know the height, and you are assuming its an isoceles trapezoid? You only have the one side of a right triangle, thats the height of 8, so i'm highly confused how people leap to the square root of three over three. Thanks, 
June 30th, 2013, 10:13 AM  #17  
Newbie Joined: Jun 2013 Posts: 2 Thanks: 0  Re: Quote:
 

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