Explorations of the Centroid of a Triangle

Kristin Karas

The centroid of a triangle is the point of concurrency when the medians are constructed. A median is the segment that extends from a vertex to the midpoint of the opposite side.

Here is a picture of what the centroid looks like.

A, B, and C are the vertices of the triangle.

M, M2, and M3 are the midpoints of the sides of the triangle.

G is the Centroid of the triangle.

Exploration: For further investigation, let's see where the centroid lies for acute, obtuse, and right triangles.

Acute Triangle: An Acute triangle has angles that are all less then 90 degrees.

**Notice that the centroid lies inside the triangle.

Obtuse Triangle: An Obtuse triangle has one angle that is greater than 90 degrees and two angles that are acute.

**Centroid remains in the interior of the triangle.

Right Triangle: A Right triangle has one right angle.

**Centroid is still in the interior of the triangle.

From this investigation, we can infer that a centroid will always be in the interior of a triangle.

Notice that when the centroid is constructed, there are 6 triangle that form with it inside the triangle. Let's investigate the area of these 6 triangles.

The area of a triangle is 1/2(b*h).

To find the area of each triangle, the perpendicular is constructed to find the height. This is then multiplied by the base and divided by two.

Here is an example for one of the triangles:

Let's repeat the process one more time:

Notice that the areas of these two triangles are both 3.99 cm squared.

In fact, the area of triangle CGM3=the area of triangle BM3G=the area of triangle BGM2= the area of triangle AGM2= 5.19 cm squared. Look below for the rest of the results.

From this investigation, we can see that the areas of the triangles formed by the centroid are equal.

We can further state that this would be true for any triangle that we construct the centroid for.

Let's look at the medial triangle. This is the triangle that is formed when we connect the midpoints of the original triangle.

Look at the measurements below:

The segments formed are half of the length of the side they are parallel to. These segments are called midsegments.

Something that we can discover from this is that the four triangles formed inside the initial triangle are congruent.

If we continue this process for the other midsegments, we can see that the triangles inside must be congruent by side-side-side-congruence postulate.