For a trapezoid, the median (also known as the midline or mid-segment) is the segment connecting the midpoints of the legs of the trapezoid. This post presents a proof that:

- the median of a trapezoid is parallel to the bases;
- the length of the median is half the sum of the lengths of the bases;
- the midpoints of the diagonals of a trapezoid also lie on its midline.

The key to all of these is the triangle midline theorem.

Consider the trapezoid *ABCD* with bases *AB* and *CD*. Let *M* and *N* be the midpoints of the legs *AD* and *BC*, respectively, and let *M′* be the midpoint of the diagonal *BD*.

Then, by the midline theorem with regards to ∆*ABD*, , and *MM′*=½*AB*. Applying the midline theorem to ∆*BCD*, , and *M′N*=½*CD*.

Now, since , we see .

Since and are both parallel to the same line , and pass through the same point *M′*, then we see that by Playfair’s axiom, they must be the same line; *M*, *M′* and *N* are collinear, and so the median . Further, *MN*=*MM′*+*M′N*=½*AB*+½*CD* =½(*AB*+*CD*).

Noting that we could perform the same procedure using diagonal *AC* and its midpoint *N′*, we see that the midpoints of both diagonals must lie on the median *MN*.

### Like this:

Like Loading...

*Related*

Tags: Math, Median, Midline, Monday Math, Playfair's Axiom, Trapezoid, Triangle Midline Theorem

This entry was posted on July 28, 2014 at 12:22 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

## Leave a Reply