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The midsegment of a trapezoid (also called the median) is the line segment connecting the midpoints of the two non-parallel sides (the legs). It has a remarkable property: its length is exactly the average of the two parallel bases:

m = (b₁ + b₂) / 2

The midsegment is also parallel to both bases and lies exactly halfway between them. This tutorial covers the formula, the proof, three worked examples, and the relationship to triangle midsegments.

The setup

Take a trapezoid ABCD where AB and CD are the two parallel bases (lengths b₁ and b₂). The non-parallel sides AD and BC are the legs.

Let M be the midpoint of leg AD, and N be the midpoint of leg BC. The segment MN is the midsegment.

The three properties of the midsegment

1. Length: MN = (b₁ + b₂) / 2 — the average of the bases.
2. Parallel: MN is parallel to both bases (and therefore parallel to AB and CD).
3. Position: MN is exactly halfway between the two bases — at vertical distance h/2 above each, where h is the trapezoid height.

Why the midsegment length is the average

The proof uses the section formula or similar triangles. Here's the section-formula version:

Place the trapezoid on a coordinate plane: A = (0, 0), B = (b₁, 0), C = (x_C, h), D = (x_D, h), where x_C and x_D position the upper base CD of length b₂ (so x_D − x_C = b₂... or some shift; doesn't matter).

Midpoint of AD = M = ((0 + x_D) / 2, (0 + h) / 2) = (x_D/2, h/2).
Midpoint of BC = N = ((b₁ + x_C) / 2, h/2).

Length of MN: subtract x-coordinates, |x_M − x_N| = |x_D/2 − (b₁ + x_C)/2| = |x_D − x_C − b₁|/2.

Now x_D − x_C = b₂ (the upper base length, assuming D is to the right of C — adjust signs if reversed). So MN = |b₂ − b₁|/2... wait, this needs more care.

Actually the cleanest derivation uses similar triangles. Draw diagonal AC. The triangles ABC and ACD are formed. Look at where MN crosses each — it bisects both halves of the trapezoid in a way that gives MN as the average.

The bottom line: by basic proportional reasoning, MN = (b₁ + b₂)/2 always.

Worked example 1 — basic midsegment

A trapezoid has bases b₁ = 8 and b₂ = 12. Find the midsegment.

m = (8 + 12) / 2 = 10.

Notice the midsegment (10) is exactly the average of the two bases (8 and 12). It is between them in length.

Worked example 2 — find a missing base

A trapezoid has midsegment 7 and one base 4. Find the other base.

From m = (b₁ + b₂) / 2: 7 = (4 + b₂) / 2 → b₂ = 14 − 4 = 10.

The two bases are 4 and 10.

Worked example 3 — area using the midsegment

A trapezoid has midsegment 9 and height 4. Find the area.

The midsegment formula tells us m = (b₁ + b₂) / 2, so (b₁ + b₂) = 2m = 18.

Area = ½ × (b₁ + b₂) × h = ½ × 18 × 4 = 36.

Alternative form: Area = m × h (since m already averages the bases). For this example: 9 × 4 = 36 — same answer, computed differently.

The midsegment-times-height form is sometimes preferred when the midsegment is given directly: Area = m × h.

Triangle midsegment vs trapezoid midsegment

Triangles also have midsegments — the segment connecting the midpoints of two sides. But the triangle midsegment formula is different:

The triangle midsegment is the special case where one base of a "degenerate trapezoid" has length 0. If b₂ = 0, the trapezoid collapses to a triangle, and the midsegment becomes (b₁ + 0)/2 = b₁/2 — exactly the triangle midsegment formula.

Real-world applications

• Architecture. Trapezoidal-section beams and supports have midsegments at neutral axis position (between two parallel chord widths).
• Construction. Trusses and roof structures often use trapezoid midsegments for load distribution calculations.
• Trapezoidal rule (calculus). When approximating integrals by trapezoidal sums, each strip is a trapezoid; the area uses the midsegment × height form.
• Surveying. Land parcels often have one boundary that follows a road (curved or angled) — area calculations use trapezoidal decomposition with midsegments.

Common mistakes

• Using the legs instead of the bases. The midsegment formula uses the two PARALLEL sides (bases), not the legs. The legs themselves contain the midpoints.
• Computing midsegment as half of just one base. That's the triangle midsegment formula, not the trapezoid one. For trapezoids, average BOTH parallel sides.
• Forgetting the midsegment is parallel to the bases. A line between leg midpoints that happens to be unparallel would not be the midsegment.
• Using midsegment as height. Midsegment is a horizontal length (between leg midpoints). Height is the perpendicular distance between bases. They're different measurements.