In this explainer, we will learn how to identify medians of a triangle and use their properties of proportionality to find a missing length.

The medians of triangles are special lines with special properties.

### Definition: Median

The medians of a triangle are the three segments going from each vertex to the midpoint of the opposite side.

The three medians intersect at a single point: we say that they are concurrent. The point of concurrency of the three medians of a triangle is called the centroid. The centroid is always inside the triangle.

### Centroid Theorem

The distance from each vertex to the centroid is two-thirds of the length of the median from this vertex.

### How To: Visualizing the Position of the Centroid

The centroid theorem means that one can split each median in three thirds as shown in the diagram. Two thirds are between the vertex and the centroid, and one third is between the centroid and the midpoint of the side.

It means that the segment between the vertex and the centroid is twice that between the centroid and the midpoint of the opposite side.

This can be found using algebra as well: If then, from the second equation (by multiplying both sides by 3), we get

By substituting with into the first equation, we get which gives

### Example 1: Knowing the Centroid Theorem

In a triangle , is the point of concurrency of its medians. If is a median, then .

### Answer

Here, the centroid (the point of concurrency of the medians) is , the vertex is , and is one of the medians of the triangle . We know from the centroid theorem that the distance from each vertex to the centroid is two-thirds of the length of the median from this vertex. It means that we have here

It means that if we cut in three equal segments, is made of two of them. It follows that is made of the third one. Therefore, is twice as long as . Hence,

### Example 2: Identifying Special Lines in a Triangle

What is the length of ?

### Answer

We derive from the diagram that is a median as it intersects at its midpoint. The same applies for , which intersects at its midpoint. passes through the intersection point of with . We deduce that is the third median since we know that the three medians are concurrent.

This means that intersects at its midpoint. The length of is therefore half of that of :

### Example 3: Applying the Centroid Theorem given the Length of a Median

Find the length of , given that .

### Answer

We derive from the diagram that both and are medians of triangle . Point is thus the centroid of triangle , that is, the point of concurrency of its medians.

We know from the centroid theorem that the length of is two-thirds that of the median :

### Example 4: Applying the Centroid Theorem given the Distance from a Vertex to the Centroid

In , . Find the length of .

### Answer

We derive from the diagram that , , and are the three medians of the triangle , and is its centroid.

We know from the centroid theorem that is two-thirds of the median , and is one-third of the median :

It follows that the length of is double that of :

So

### Example 5: Applying the Centroid Theorem with Algebra

In , and . Find .

### Answer

In triangle , is the centroid (the point of concurrence of its medians). is a vertex and is the middle of the opposite side, , therefore, according to the centroid theorem, , from which it follows that , and so . It is said in the question that and , hence, we have

Dividing both sides of this equation by 2 gives

Adding 7 to both sides and then dividing both sides by 5 gives

### Key Points

- The medians of a triangle are the three segments going from each vertex to the midpoint of the opposite side.
- The three medians intersect at a single point: we say that they are concurrent. The point of concurrency of the three medians of a triangle is called the centroid. The centroid is always inside the triangle.
- The centroid theorem states that the distance from each vertex to the centroid is two-thirds of the length of the median from this vertex.
- The centroid theorem means that one can split each median in three thirds as shown in the diagram. Two thirds are between the vertex and the centroid, and one third is between the centroid and the midpoint of the side. It means that the segment between the vertex and the centroid is twice that between the centroid and the midpoint of the opposite side.