Detailed definition
Understanding Median
Median is a segment from a vertex to the midpoint of the opposite side. A median connects a vertex to the midpoint of the opposite side. Since each triangle has three vertices, it also has three medians.
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.
Key facts
Important ideas to remember
- 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 it is used
Where median shows up
- 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.
Common mistakes
What to watch out for
- 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.