GeoProof — Trapezoid Midsegment explorer

MN = (AB + DC) / 2

Trapezoid Midsegment

The midsegment (or median) of a trapezoid joins the midpoints of its two non-parallel sides (the legs). It is parallel to both bases, and:

MN = (AB + DC) / 2

Its length is simply the average of the two parallel sides.

Why it averages

Writing the legs as vectors, the midsegment vector is the average of the two base vectors: N − M = ½((B − A) + (C − D)). Since both bases point the same way, this is parallel to them and its length is the mean of their lengths.

Two familiar limits

Shrink the top base to a point and the trapezoid becomes a triangle — the midsegment is then half the remaining base, the triangle midsegment theorem. Make both bases equal and you get a parallelogram, where the midsegment matches each base.

Where it's used

It gives the trapezoid area as midsegment × height, and underlies the trapezoidal rule for estimating areas under curves.