The Triangle Midsegment Theorem
Prove that the midsegment of a triangle is parallel to the third side and half its length. Apply the theorem to find missing lengths and use coordinate proof to verify the theorem algebraically.
A lesson of Geometry, in Triangles and Congruence, about 10 minutes.
By the end of this lesson, you will be able to identify midsegments of triangles, apply the Triangle Midsegment Theorem (midsegment is parallel to the base and half its length), and use coordinates to verify midsegment properties.
The Big Idea
A midsegment creates a smaller triangle inside a larger one, with two beautiful properties: it is parallel to the third side and exactly half its length. This theorem appears in proofs, coordinate problems, and real-world applications from architecture to computer graphics.
Definition
A midsegment of a triangle is a segment connecting the midpoints of two sides of the triangle.
Every triangle has three midsegments (one for each pair of sides).
Triangle Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle is:
- Parallel to the third side
- Half as long as the third side
If $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{AC}$ respectively in $\triangle ABC$:
$$\overline{DE} \parallel \overline{BC} \quad \text{and} \quad DE = \frac{1}{2}BC$$
Proof Using Coordinates
Place $\triangle ABC$ with $A = (0, 0)$, $B = (2b, 0)$, $C = (2c, 2d)$.
Midpoint of $\overline{AB}$: $D = (b, 0)$
Midpoint of $\overline{AC}$: $E = (c, d)$
Slope of $\overline{DE}$: $\dfrac{d - 0}{c - b} = \dfrac{d}{c-b}$
Slope of $\overline{BC}$: $\dfrac{2d - 0}{2c - 2b} = \dfrac{d}{c-b}$
Equal slopes → DE ∥ BC ✓
$DE = \sqrt{(c-b)^2 + d^2}$
$BC = \sqrt{(2c-2b)^2 + (2d)^2} = 2\sqrt{(c-b)^2 + d^2}$
$DE = \dfrac{1}{2} BC$ ✓
Example 1 — Finding Midsegment Length
In $\triangle ABC$, $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{BC}$. If $AC = 18$, find $DE$.
By the Midsegment Theorem: $DE = \dfrac{1}{2}(18) = 9$
Example 2 — Finding the Third Side
Midsegment $\overline{MN} = 13$. Find the length of the parallel side.
$BC = 2 \times 13 = 26$
Example 3 — Algebra
$D$ and $E$ are midpoints of two sides of a triangle. $DE = 3x + 2$ and $BC = 5x + 10$. Find $DE$.
$$3x + 2 = \frac{1}{2}(5x + 10) \implies 3x + 2 = \frac{5x}{2} + 5$$
$$6x + 4 = 5x + 10 \implies x = 6$$
$DE = 3(6) + 2 = 20$
Example 4 — Coordinate Geometry
$\triangle PQR$ has $P(0, 0)$, $Q(6, 0)$, $R(4, 8)$. Find the midsegment connecting the midpoints of $\overline{PQ}$ and $\overline{PR}$, verify it's parallel to $\overline{QR}$ and half its length.