What is a median in a triangle?

How the segment is identified

A median is not defined by perpendicularity. Its defining condition is the midpoint on the opposite side. In some special triangles a median may also be an altitude or angle bisector, but that is extra information, not the basic definition.

Medians matter because they divide the triangle into equal-area pairs and all three meet at the centroid. That makes median one of the most useful structural segments in triangle geometry.

If a side has endpoints at two marked points, first find the point halfway between them and then connect that midpoint to the opposite vertex. The segment is a median because of that halfway endpoint, even if it is not perpendicular and even if the triangle is tilted.

A median divides the triangle into two regions with equal area because the two smaller triangles share the same altitude to the opposite side and have equal bases. That area fact is a consequence of the midpoint, not a second definition that must be checked separately.

• A median connects a vertex to the midpoint of the opposite side.
• A median must end at the midpoint of the opposite side.
• Each median divides the triangle into two smaller triangles of equal area.
• The three medians intersect at the centroid.

Where medians help

• Use medians in centroid and balance-point problems.
• Use them when a triangle diagram marks a midpoint on one side.
• Use median facts in proofs that depend on equal-area subdivisions.

Three median checks

• Do not call a segment a median just because it starts at a vertex; the endpoint must be the midpoint of the opposite side.
• Do not confuse midpoint structure with perpendicular structure.
• Do not assume every median is also an altitude unless the triangle's symmetry shows that extra fact.