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-   -   how to show that two sides of a trapezoid are parallel? (http://mymathforum.com/geometry/332269-how-show-two-sides-trapezoid-parallel.html)

davedave May 30th, 2016 02:49 PM

how to show that two sides of a trapezoid are parallel?
 
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Attachment 7620

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 x-axis. Points B and C are above the x-axis 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.

mrtwhs May 30th, 2016 04:47 PM

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 $\displaystyle (f-c)^2=0$.

This tells you that BC and MN are both horizontal and thus parallel to AD.


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