What is the Trapezoid Midsegment?
The midsegment (also called the median or midline) of a trapezoid is the line segment connecting the midpoints of the two non-parallel sides (legs). It is parallel to both bases and its length equals the arithmetic mean (average) of the two parallel sides.
Midsegment Properties
Length Formula
The midsegment length is the average of the two parallel sides.
Parallel to Bases
The midsegment is always parallel to both bases of the trapezoid.
Area via Midsegment
The area of a trapezoid equals the midsegment times the height.
Bisects the Legs
The midsegment passes through the midpoints of both legs, dividing each leg into two equal halves.
Derivation
The midsegment theorem for trapezoids states: "The segment connecting the midpoints of the legs of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths." This can be proved using coordinate geometry or the triangle midpoint theorem applied to the diagonal triangles.
Relationship to Area
Since A = ½(a + b) × h and m = (a + b)/2, we can write A = m × h. This gives a particularly elegant way to compute the area: it equals the midsegment length times the height, similar to how a rectangle's area is width times height.
Special Cases
- If a = b, the trapezoid is actually a parallelogram and the midsegment equals both bases.
- If a = 0, the trapezoid degenerates into a triangle and the midsegment equals b/2 (the triangle midline theorem).
- The midsegment is always between a and b in length: min(a,b) ≤ m ≤ max(a,b).