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′=½AB. Applying the midline theorem to ∆BCD, , and M′N=½CD.
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=½AB+½CD =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.