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Median of a Triangle — Definition, Formula & Examples

Median of a Triangle

A line segment drawn from one vertex to the midpoint of the opposite side. Note: The three medians of a triangle are concurrent, intersecting at the centroid.

 

Triangle with one median drawn from a vertex to the midpoint of the opposite side, labeled "median.

 

 

See also

Centers of a triangle, median of a set of numbers

Key Formula

ma=122b2+2c2−a2m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}
Where:
  • mam_a = Length of the median drawn from vertex A to the midpoint of side a
  • aa = Length of the side opposite vertex A (the side the median lands on)
  • bb = Length of the side opposite vertex B
  • cc = Length of the side opposite vertex C

Worked Example

Problem: Triangle ABC has vertices at A(0, 0), B(6, 0), and C(2, 4). Find the length of the median from vertex A to the midpoint of side BC.
Step 1: Find the midpoint M of side BC. The midpoint formula gives the average of the coordinates.
M=(6+22,  0+42)=(4,  2)M = \left(\frac{6+2}{2},\; \frac{0+4}{2}\right) = (4,\; 2)
Step 2: The median from A goes from A(0, 0) to M(4, 2). Use the distance formula to find its length.
ma=(4−0)2+(2−0)2=16+4=20m_a = \sqrt{(4-0)^2 + (2-0)^2} = \sqrt{16 + 4} = \sqrt{20}
Step 3: Simplify the radical.
ma=25≈4.47m_a = 2\sqrt{5} \approx 4.47
Answer: The median from vertex A to the midpoint of BC has length 25≈4.472\sqrt{5} \approx 4.47 units.

Another Example

Problem: A triangle has sides a = 8, b = 6, and c = 10. Find the length of the median to side a using the median length formula.
Step 1: Identify which median you need. The median to side a uses the formula with a, b, and c.
ma=122b2+2c2−a2m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}
Step 2: Substitute the given side lengths into the formula.
ma=122(6)2+2(10)2−(8)2=1272+200−64m_a = \frac{1}{2}\sqrt{2(6)^2 + 2(10)^2 - (8)^2} = \frac{1}{2}\sqrt{72 + 200 - 64}
Step 3: Simplify under the radical and compute.
ma=12208=12⋅413=213≈7.21m_a = \frac{1}{2}\sqrt{208} = \frac{1}{2} \cdot 4\sqrt{13} = 2\sqrt{13} \approx 7.21
Answer: The median to side a has length 213≈7.212\sqrt{13} \approx 7.21 units.

Frequently Asked Questions

Where do the three medians of a triangle intersect?
The three medians always intersect at exactly one point called the centroid. The centroid divides each median in a 2:1 ratio, measured from the vertex to the midpoint of the opposite side. This means the centroid is located two-thirds of the way along each median from its vertex.
Does every triangle have exactly three medians?
Yes. Since a triangle has three vertices and three opposite sides, there is exactly one median from each vertex to the midpoint of the opposite side, giving three medians in total. This is true for all triangles — scalene, isosceles, and equilateral.

Median of a Triangle vs. Altitude of a Triangle

A median connects a vertex to the midpoint of the opposite side, while an altitude connects a vertex to the opposite side (or its extension) at a right angle. The median bisects the opposite side; the altitude is perpendicular to it. The three medians meet at the centroid, whereas the three altitudes meet at the orthocenter.

Why It Matters

Medians are essential for locating the centroid, which is the balance point (center of mass) of a triangle. If you cut a triangle out of uniform cardboard, it would balance perfectly on the centroid. Medians also appear in coordinate geometry proofs and are used in physics and engineering to find centers of gravity of triangular regions.

Common Mistakes

Mistake: Confusing a median with an altitude by drawing a perpendicular line to the opposite side instead of connecting to its midpoint.
Correction: A median must end at the midpoint of the opposite side, regardless of the angle it makes. An altitude must be perpendicular to the opposite side, regardless of where it lands. These are the same segment only in special cases like an equilateral triangle or the median from the apex of an isosceles triangle.
Mistake: Thinking the centroid divides each median into two equal halves.
Correction: The centroid divides each median in a 2:1 ratio, not 1:1. The longer segment (from vertex to centroid) is twice the length of the shorter segment (from centroid to midpoint of the side).

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