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Investigations with
Centroids

By: Lauren Lee

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The **centroid** is the
point of intersection of the medians of a triangle.

Medians are found by constructing segments from the vertices of the triangle to the midpoints of the sides opposite the vertices. So there will always be three medians in a triangle.

Let’s look at one example of a median:

For our triangle ADC, B is the midpoint of the side AC. So DB is one median of the triangle.

Now let’s look at the centroid of a triangle.

To do this, we will first have to construct the three medians of the triangle and then find the point of intersection.

In this example our medians are AE, BF, and CD. The point of intersection of the medians,
which is our **centroid**, is labeled G
because it is the center of gravity of the triangle.

Now that we know how to find the centroid, let’s investigate! Will the centroid always be inside the triangle?

Let’s look at three types of triangles: right triangle, acute triangle, and obtuse triangle.

Right Triangle

Acute Triangle

Obtuse Triangle

The centroid G is always inside the triangle, regardless the type, because it is the center of gravity.

Let’s
look at the medians of a triangle once more.

Here’s an example of a triangle with the distances between two points of a median noted below.

We can see that AG is twice as long as GE. Similarly, BG = 2GF and CG = 2GD.

So our
final observation is that the medians of a triangle intersect at a point that
is two thirds of the distance from each vertex to the midpoint of the opposite
side. In other words, the longer
segment of a median is always twice as long as the shorter segment.