Similarity & Trigonometry
GradesGeometry & FunctionsSimilarity & TrigonometryTriangle Midsegment Theorem

Triangle Midsegment Theorem

📐 Similarity & Trigonometry
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Triangle Midsegment Theorem

Connect midpoints to reveal a segment parallel to and half the length of the third side.

The Triangle Midsegment Theorem states: a segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.

If M=midpoint of AB and N=midpoint of AC, then MN∥BC and MN=12BC\text{If } M = \text{midpoint of } AB \text{ and } N = \text{midpoint of } AC,\text{ then } MN \parallel BC \text{ and } MN = \frac{1}{2}BC
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The midsegment is parallel to the base AND exactly half its length — both facts hold simultaneously.

✏️Using the Midsegment Theorem
In △ABC, DE is the midsegment connecting the midpoints of AB and AC. If BC = 18, then DE = 9 (half of 18). Also ∠ADE = ∠ABC (corresponding angles — DE ∥ BC).

Applying the Theorem in Coordinate Geometry

1Find the midpoints M and N of two sides using the midpoint formula.
2Compute the length of MN using the distance formula.
3Compute the length of the third side (base) and verify MN = base/2.
4Check slopes of MN and the base are equal to confirm parallelism.
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Remember This!

Every triangle has three midsegments — one connecting each pair of side midpoints. Together they form the medial triangle, which divides the original triangle into 4 congruent smaller triangles.

✏️ Try It!

In △PQR, the midsegment ST connects midpoints of PQ and PR. If ST = 13, what is QR?

Take Quiz 📝 — 25 Questions