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Centroid

Math is filled with all kinds of interesting words, and "centroid" is definitely up there with the coolest-sounding names. But what exactly is a centroid? As we'll soon discover, a centroid is a point that is related to the "middle" of the triangle and something we can always find. Even better, we can calculate the centroid of a triangle using a relatively straightforward formula. Let's dive into this topic:

What is the centroid?

Every triangle has a centroid. As the name implies, this point is located right in the center of each triangle. But in order to understand centroids, we need to know about medians. The median of a triangle is a straight line that goes from an apex to the midpoint of the opposite side.

Why do we need to know about medians if we're learning about centroids? Simple: We need to put our centroid right at the meeting place between the three medians. Each triangle has three medians, which means we need to draw all of these lines before we know the exact location of the centroid.

Here's what a triangle looks like when we draw all three medians. As you can see, there's a clear point in the middle where all three lines intersect:

We can also see that this triangle has a few interesting properties:

• The median divides the triangle into two equal parts with the same areas
• The three medians always meet at a single centroid
• The three medians divide the triangle into six smaller triangles that each have identical areas

The centroid theorem

You've heard about the Pythagorean theorem.. But what about the centroid theorem?

This formula helps us determine the distance from the centroid to the midpoint of a side. It also helps us find out the distance from the centroid to an apex. The theorem is very simple:

The centroid is always $\frac{2}{3}$ of the distance from the vertex to the midpoint of the corresponding side along a median. We could also flip this upside down and say that the centroid is always $\frac{1}{3}$ of the distance from the midpoint of the side to the apex.

Take another look at this triangle:

Based on this triangle, we can turn the centroid theorem into the following equations:

$\mathrm{QV}=\frac{2}{3}\mathrm{QU}$

$\mathrm{PV}=\frac{2}{3}\mathrm{PT}$

$\mathrm{RV}=\frac{2}{3}\mathrm{RS}$

Thought we were done with centroid formulas? Think again!

There's another interesting formula we can try out if we know the coordinates of a triangle on the Cartesian plane. As you might remember, the Cartesian plane has four quadrants, and it allows us to translate geometric shapes into algebraic equations. Cartesian coordinates take the form of ordered pairs and can be either positive or negative.

Under this centroid formula, a triangle would have the following coordinates:

A at $\left({x}_{1},{y}_{1}\right)$ , B at $\left({x}_{2},{y}_{2}\right)$ , and C at $\left({x}_{3},{y}_{3}\right)$

To find the centroid of a triangle based on these coordinates, all we need to do is find the average of x and y coordinates of all three vertices. We can follow this equation to make this calculation:

$\mathrm{Centroid}=\left(\frac{{x}_{1}+{x}_{2}+{x}_{3}}{3},\frac{{y}_{1}+{y}_{2}+{y}_{3}}{3}\right)$

Example:

Let's assume that a triangle has the following coordinates:

$A\left(2,6\right)$ , $B\left(4,9\right)$ , and $C\left(6,15\right)$

Our formula would look like this:

$\left(\frac{2+4+6}{3},\frac{6+9+15}{3}\right)$

This would give us $\left(\frac{12}{3},\frac{30}{3}\right)$

If we simplify these coordinates, we know that the centroid is $\left(4,10\right)$

Don't worry if you get coordinates that are fractions or negative numbers. Remember, negative coordinates and fractions are completely acceptable on the Cartesian plane.

Why is the centroid important?

Finding the centroid may be useful for a number of real-world problems. The centroid always represents an object's center of gravity (assuming the objects has a uniform density), and it can apply to a number of shapes as well as triangles. This means that if you were to spin a triangle on its exact centroid, it wouldn't topple over.