Constructing the Centroid - Concept
The centroid is the point of concurrency of the three medians in a triangle. It is the center of mass (center of gravity) and therefore is always located within the triangle. The centroid divides each median into a piece one-third the length of the median and two-thirds the length. To find the centroid, we find the midpoint of two sides in the coordinate plane and use the corresponding vertices to get equations.
One of the four main types of points of concurrency in a triangle is the Centroid. Couple of key things about the centroid.
The first is that it's where the three medians intersect each other. That's a definition of a point of concurrence. It is also the center of mass or I guess you could call the center of gravity. But since your mass doesn't change no matter where you are, it's a little more technical to say the center of mass. It also divides each segment into two proportional pieces, one two thirds of the length and one being one third of the length. Well, that's a little confusing.
So let's take a look at a specific example. So here I've drawn in the medians and again the median is the line segment from a vertex to the midpoint of the opposite side. So I'm going to mark that. We have congruent pieces here to show you that we have three medians and they're concurrent.
And so the first thing about the center of mass is if I actually built this triangle on a piece of construction paper or cardboard, the place that would balance this triangle is right at the centroid. So I could flip it upside down and I could balance it on my marker here if I have constructed that point.
Getting back to the proportional part. If I looked at this median be so I'm going to re-draw. Down below I'm going to have b up here and I'm going to have e down there. The median is right about here. So of this whole line segment two thirds of it will be between I guess I could call this point f. Two thirds will be between b and f and one third will be between f and e. So if this whole segment was let's say 12 centimetres then you could say that this part right here bf would be eight centimetres and fe would be 4 centimetres.
So the centroid is a point of concurrency of the three medians and it's also known as the center of mass.