Midsegments & Medians

A midsegment is half the third side; the centroid splits a median 2:1.

The idea

The midsegments and medians of a triangle are the segments determined by the midpoints of its sides.

A midsegment joins the midpoints of two sides: in triangle $ABC$, with $M$ and $N$ the midpoints of $AB$ and $AC$, the segment $MN$ is a midsegment.

Theorem (Midsegment Theorem).

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

Every triangle has three midsegments, one for each pair of sides; drawing all three cuts the triangle into four smaller triangles.

A median joins a vertex to the midpoint of the opposite side, so every triangle has three medians, one from each vertex.

Theorem (Medians and the centroid).

The three medians of a triangle pass through a single point, the centroid, which divides each median in the ratio $2 : 1$ measured from the vertex.

The centroid is the triangle's balance point. In lengths, the distance from a vertex to the centroid is $\tfrac{2}{3}$ of that median's length, and the distance from the centroid to the midpoint is the remaining $\tfrac{1}{3}$.

Ways to work on it

• Walkthrough. Midsegment halving and the centroid's 2:1 median split.
• Proof. Why the midsegment is parallel and half: SAS similarity of AMN.
• Practice. Convert between a side and its midsegment.
• Hardest. Recover a median or a piece from the 2:1 ratio.

Not sure where to start? Take the ten-question placement test.