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May 30th, 2016, 02:49 PM  #1 
Senior Member Joined: Apr 2008 Posts: 193 Thanks: 3  how to show that two sides of a trapezoid are parallel? parallel lines.pdf Please click the link above to see a diagram. The diagram displays a trapezoid with vertices ABCD. A is at the origin and D is to the right of A on the xaxis. Points B and C are above the xaxis in the first quadrant. M is the midpoint between A and B. N is the midpoint between C and D. Let MN be a line segment between M and N. BC and AD are the top and bottom sides of the trapezoid. Given MN = 0.5(BC + AD), prove that BC is parallel to AD. This problem is awfully hard. Can someone please help me? Thank you very much. 
May 30th, 2016, 04:47 PM  #2 
Senior Member Joined: Feb 2010 Posts: 679 Thanks: 127 
Coordinatize everything: A(0,0) B(2b,2c) C(2e,2f) D(2d,0) The midpoints are M(b,c) and N(d+e,f) Start with 2MN = BC + AD, use the standard distance formula, plug in, square twice (cancelling on the way). Things cancel and simplify very quickly and you end up with $\displaystyle (fc)^2=0$. This tells you that BC and MN are both horizontal and thus parallel to AD. 

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parallel, show, sides, trapezoid 
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