The Triangle Midsegment Theorem

A midsegment joins the midpoints of two sides of a triangle, and is parallel to the third side with half its length. Learn the theorem, its converse, and how to use it.

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A midsegment of a triangle is the segment that joins the midpoints of two of its sides. It has a simple, powerful relationship with the third side.

Theorem The Midsegment Theorem

The segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is half of it.

A B C M N

M and N are the midpoints of AB and AC, so:

MN \parallel BC, \quad MN = \tfrac{1}{2}\,BC

Note The converse

It also works the other way: a segment drawn inside the triangle parallel to one side, with half its length, joins the midpoints of the other two sides — so it splits them exactly in half.

Example Using the midsegment

In triangle RST, X and Z are the midpoints of RS and RT, so XZ is a midsegment.

If ST = 13, then XZ = \tfrac{1}{2}\times 13 = 6.5 .
If instead XZ = 7, then ST = 2 \times 7 = 14 .
⟹ the midsegment is always half the third side.
Summary
  1. A midsegment joins the midpoints of two sides of a triangle.
  2. It is parallel to the third side and exactly half its length.
  3. Because it is parallel, alternate interior angles with the third side are equal — useful for finding angles.
  4. The converse is true too: a parallel segment of half the length meets the other two sides at their midpoints.
231. Arithmetic Sequences and Series
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