Explainer: Medians of Triangles

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 Visualize 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 𝐴𝑃=23⋅𝐴𝐸𝑃𝐸=13⋅𝐴𝐸,and then, from the second equation (by multiplying both sides by 3), we get 𝐴𝐸=3𝑃𝐸.

By substituting 𝐴𝐸 with 3𝑃𝐸 into the first equation, we get 𝐴𝑃=23β‹…3𝑃𝐸 which gives 𝐴𝑃=2𝑃𝐸.

Example 1: Knowing the Centroid Theorem

In a triangle 𝐴𝐡𝐢, 𝑀 is the point of concurrency of its medians. If 𝐴𝐷 is a median, then 𝐴𝑀=𝑀𝐷.

Answer

Here, the centroid (the point of concurrency of the medians) is 𝑀, the vertex is 𝐴, and 𝐴𝐷 is one of the medians of the triangle 𝐴𝐡𝐢. 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 𝐴𝑀=23𝐴𝐷.

It means that if we cut 𝐴𝐷 in three equal segments, 𝐴𝑀 is made of two of them. It follows that 𝑀𝐷 is made of the third one. Therefore, 𝐴𝑀 is twice as long as 𝑀𝐷. Hence, 𝐴𝑀=2𝑀𝐷.

Example 2: Identifying Special Lines in a Triangle

What is the length of 𝐢𝐷?

Answer

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

This means that 𝐡𝐷 intersects 𝐴𝐢 at its midpoint. The length of 𝐢𝐷 is therefore half of that of 𝐴𝐢: 𝐢𝐷=12⋅𝐴𝐢=12β‹…13.8=6.9.cm

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

Find the length of 𝐴𝑀, given that 𝐴𝐸=54.

Answer

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

We know from the centroid theorem that the length of 𝐴𝑀 is two-thirds that of the median 𝐴𝐸: 𝐴𝑀=23⋅𝐴𝐸=23β‹…54=36.

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

In △𝐽𝐾𝐿, 𝐽𝑃=6cm. Find the length of 𝑃𝑆.

Answer

We derive from the diagram that 𝐾𝑇, 𝐿𝑅, and 𝐽𝑆 are the three medians of the triangle 𝐾𝐿𝐽, and 𝑃 is its centroid.

We know from the centroid theorem that 𝐽𝑃 is two-thirds of the median 𝐽𝑆, and 𝑃𝑆 is one-third of the median 𝐽𝑆: 𝐽𝑃=23⋅𝐽𝑆;𝑃𝑆=13⋅𝐽𝑆.

It follows that the length of 𝐽𝑃 is double that of 𝑃𝑆: 𝐽𝑃=2⋅𝑃𝑆.

So 𝑃𝑆=12⋅𝐽𝑃=12β‹…6=3.cm

Example 5: Applying the Centroid Theorem with Algebra

In △𝐾𝑀𝐻, 𝐾𝑄=2 and 𝑄𝑃=(5π‘₯βˆ’7). Find π‘₯.

Answer

In triangle 𝐾𝑀𝐻, 𝑄 is the centroid (the point of concurrence of its medians). 𝐾 is a vertex and 𝑃 is the middle of the opposite side, 𝐻𝑀, therefore, according to the centroid theorem, 𝐾𝑄=23𝐾𝑃, from which it follows that 𝑄𝑃=13𝐾𝑃, and so 𝐾𝑄=2𝑄𝑃. It is said in the question that 𝐾𝑄=2 and 𝑄𝑃=5π‘₯βˆ’7, hence, we have 2=2(5π‘₯βˆ’7).

Dividing both sides of this equation by 2 gives 1=5π‘₯βˆ’7.

Adding 7 to both sides and then dividing both sides by 5 gives π‘₯=85=1.6.

Key Points

  1. The medians of a triangle are the three segments going from each vertex to the midpoint of the opposite side.
  2. 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.
  3. 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.
  4. 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.

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