Median of a Triangle: Properties & Formula
Median of a Triangle: Properties & Formula
What Is the Median of a Triangle?
The median of a triangle is a line segment that joins a vertex to the midpoint of the side opposite that vertex. Because it always ends at the midpoint, a median bisects the opposite side into two equal pieces.
Every triangle has exactly three medians, one drawn from each of its three vertices. Unlike an altitude, a median is defined by a point (the midpoint), not by an angle, so it does not have to meet the opposite side at a right angle. All three medians always lie inside the triangle, whatever its shape.
Properties of the Median of a Triangle
The join-to-the-midpoint rule forces a clean and surprisingly rich set of properties:
Three medians, all inside. Every triangle has three, and none ever falls outside the figure.
Each median bisects its side. It splits the opposite side into two equal lengths.
Each median halves the area. A single median divides the triangle into two smaller triangles of equal area, because they share the same height and have equal bases.
The three medians meet at the centroid. This single common point exists for every triangle.
The three medians cut the triangle into six equal-area pieces. Together they slice it into six small triangles, all of the same area.
What Is the Centroid (and the 2:1 Rule)?
The three medians of any triangle always meet at one point called the centroid (often written $G$), and it is the triangle's centre of mass, the balance point from the hook above.
The centroid does something precise to every median. It divides each one in the ratio 2 : 1, measured from the vertex:
$$\text{vertex-to-centroid} : \text{centroid-to-midpoint} = 2 : 1.$$
So the centroid sits two-thirds of the way along every median, counting from the vertex. If a median is 9 cm long, the centroid is 6 cm from the vertex and 3 cm from the midpoint.
For a triangle placed on coordinates, the centroid is just the average of the three vertices:
$$G = \left( \frac{x_1 + x_2 + x_3}{3}, ; \frac{y_1 + y_2 + y_3}{3} \right).$$
The Length of a Median (Apollonius's Theorem)
The length follows from Apollonius's theorem, which relates a median to the three side lengths. For a triangle with sides $a$, $b$, $c$, the median $m_a$ drawn to side $a$ has length:
$$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}.$$
Median vs Altitude: What Is the Difference?
| Feature | Median | Altitude |
|---|---|---|
| Goes from a vertex to | the midpoint of the opposite side | the opposite side, at 90° |
| Always bisects the base? | Yes, by definition | No |
| Always perpendicular? | No, not usually | Yes |
| Stays inside the triangle? | Yes, always | No (outside for obtuse) |
| Three of them meet at | the centroid | the orthocentre |
They coincide only in symmetric cases, such as the median from the apex of an isosceles triangle, which is also the altitude to the base.
Examples of Median of a Triangle
Example 1 - A median of a triangle is 12 cm long. How far is the centroid from the vertex, and from the midpoint?
Final answer: 8 cm from the vertex, 4 cm from the midpoint.
Example 2 - The centroid of a triangle is 10 cm from a vertex along one median. Find the full length of that median
Final answer: 15 cm.
Example 3 - In triangle ABC, sides are $b = 6$ cm, $c = 8$ cm, and $a = 10$ cm. Find the length of the median to side $a$
Final answer: 5 cm.
Example 4 - A triangle has vertices $A(4, 10)$, $B(8, 2)$, and $C(-8, 4)$. Find the centroid
Final answer: $G = \left(\tfrac{4}{3}, \tfrac{16}{3}\right) \approx (1.33, 5.33)$.
Example 5 - Find the median to the longest side of a triangle with sides 5 cm, 7 cm, and 8 cm
Final answer: about 4.58 cm.
Example 6 - An equilateral triangle has side 6 cm. Find the length of any median
Final answer: $3\sqrt{3} \approx 5.20$ cm.
Why the Median of a Triangle Matters
The median is more than a textbook line, it is the geometry of balance and the structure behind several deeper results.
Centre of mass. The centroid is where a flat triangular object balances and where its weight effectively acts.
The Euler line and triangle centres. The centroid sits with the orthocentre and circumcentre on a single straight line, the Euler line.
Equal-area division. Because a median splits a triangle into two equal areas, it is the natural tool for fairly dividing a triangular region.
Where Students Trip Up on Medians
Mistake 1: Reading the 2:1 ratio as "halfway"
The correct way: The centroid is two-thirds of the way from the vertex, a 2:1 split.
Mistake 2: Confusing the median with the altitude
The correct way: A median goes to the midpoint but is usually not perpendicular; an altitude is perpendicular.
Mistake 3: Mis-assigning the sides in Apollonius's theorem
The correct way: The side the median lands on (the one it bisects) is the one subtracted.
Key Takeaways
The median of a triangle joins a vertex to the midpoint of the opposite side, bisecting it; every triangle has three.
A single median splits the triangle into two equal areas; the three together make six equal-area pieces.
The three medians meet at the centroid, which divides each median 2:1 from the vertex.
The median's length comes from Apollonius's theorem, $m_a = \tfrac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$.