Proof: Median of a Trapezoid Theorem

Date: 03/24/2001 at 18:28:25
From: Heather Miller
Subject: 10th grade geometry proofs

I was wondering if you could tell me the answer to a two-column proof.

It is as follows: 

The Median of a Trapezoid Theorem: the median of a trapezoid is: 
  1) parallel to its bases
  2) length = 1/2 the sum of the bases

Can you help me prove that?

Thank you,
Heather

Date: 03/25/2001 at 03:37:52
From: Doctor Floor
Subject: Re: 10th grade geometry proofs

Hi, Heather,

Thanks for writing.

There is a similar theorem for triangles, of which you can find a 
proof in the Dr. Math archives at:

  Mid-segment Theorem
  http://mathforum.org/dr.math/problems/richard02.02.99.html   

Now you can apply this theorem in the following way:


          D---C       M, N and O midpoints of AD, DB and BC.
        _- -_  \
      _M     N_ O
    _-         -_\
   A--------------B

By the midsegment theorem, MN is paralell to MN and half its length, 
while NO is parallel to DC and half its length. 

Since DC and AB are parallel MNO are on one line.

The length of median MO is thus (AB+DC)/2, as desired.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/