Medians in Triangles

Learning Objectives

  • Construct the medians of a triangle.
  • Apply the Concurrency of Medians Theorem to identify the point of concurrency of the medians of the triangle (the centroid).
  • Use the Concurrency of Medians Theorem to solve problems involving the centroid of triangles.

Definition of Median of a Triangle

A median of a triangle is the line segment that joins a vertex to the midpoint of the opposite side.

Here is an example that shows the medians in an obtuse triangle.

That the three medians appear to intersect in a point is no coincidence. As was true with perpendicular bisectors of the triangle sides and with angle bisectors, the three medians will be concurrent (intersect in a point). We call this point the centroid of the triangle. We can prove the following theorem about centroids.

The Centroid of a Triangle

Concurrency of Medians Theorem: The medians of a triangle will intersect in a point that is two-thirds of the distance from the vertices to the midpoint of the opposite side.

Consider \triangle ABC with midpoints of the sides located at R, S, and T and the point of concurrency of the medians at the centroid, X. The theorem states that CX = \frac{2} {3} CS , AX = \frac{2}{3} AT, and BX = \frac{2} {3} BR.

The theorem can be proved using a coordinate system and the midpoint and distance formulas for line segments. We will leave the proof to you (Homework Exercise #10), but will provide an outline and helpful hints for developing the proof.

Example 1.

Use The Concurrency of Medians Theorem to find the lengths of the indicated segments in the following triangle that has medians \overline{AT}, \overline{CS}, and \overline{BR} as indicated.

1. If CS = 12, then CX = ____ and XS = ____.

CX & = \frac{2}{3}\cdot 12 = 8 \\ XS & = \frac{1}{3}\cdot 12 = 4

2. If AX = 6. then XT = ____ and AT = _____.

We will start by finding AT.

AX &= \frac{2}{3} AT \\ 6 &= \frac{2}{3} AT \\ 9 &= AT

Now for XT,

XT &= \frac{1}{3} AT \\ &= \frac{1}{3} 9 \\ &=3

Lesson Summary

In this lesson we:

  • Defined the centroid of a triangle.
  • Stated and proved the Concurrency of Medians Theorem.
  • Solved problems using the Concurrency of Medians Theorem.


Points to Consider

So far we have been looking at relationships within triangles. In later chapters we will review the area of a triangle. When we draw the medians of the triangle, six smaller triangles are created. Think about the area of these triangles, and how that might relate to example 1 above.


Last modified: Monday, June 28, 2010, 3:56 PM