# 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, $m_a = \tfrac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$.