Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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The **centroid** is the point of concurrency of three medians of a triangle. Remember that a **median** is a segment connecting a vertex to the midpoint of the opposite side. So, first construct the medians. It is only necessary to construct two because the third will intersect the others at the same point. Thus, the point of intersection of two medians is the point of concurrency, and the centroid of the triangle.

A construction problem involving points of congruency with a bigger problem like this where you are being asked to construct the centroid of this triangle. But what is a centroid?

Well the centroid, we said was a point of concurrency of a three medians. So in order to find this first we’ll have to construct the medians and then see where they intersect. We said that also is a center of mass and that it would divide each median proportionally. So let’s get back to this triangle.

If we were to kind of sketch a game plan here, I’m going to redraw this triangle over here and the first thing I’m going to do construct a median from one vertex to the midpoint of the opposite side. So I know if I construct the midpoint of this side I’ll be halfway there and then I’m going to construct another median so I’m going to have to find two midpoints and then connect them to my vertices and that will be my centroid.

So we’ve got to plan time to execute it. You're going to grab your compass and the first step is to bisect one of your sides. I’m going to choose to bisect this side so I’ll to put the sharp end of my compass to make sure that I’m more than half of this segment right here and I’m going to swing an arc from this vertex. I’m going to come up here and swing another arc from this vertex and we should have two points of intersection which we do.

So I want to know where is the midpoint. So I don’t need to draw this line but if I did it would be the perpendicular bisector, so our midpoint is right there. If I connect this midpoint to the vertex that’s called the median so I’m going to go ahead and do that. So now we have one median which has divided that side and the two congruent pieces. Let's do it one more time. grab your compass let’s try bisecting this side right here I’m going to change my compass setting a little bit and I’m going to swing an arc and I’m going to come from this other side and I’m going to realize that I didn’t swing a big enough arc here, so I’m going to extend it a little bit.

So now I can go back up here and just swing an arc from my other vertex and I’m going to find my midpoint. My midpoint is right there so I’m going to connect my midpoint with my vertex which is also called the median and at this point right here where two medians have intersected that is our centroid.

We don’t have to construct our third median because we know as a point of concurrency if two of these intersected, the third will as well. So remember the point of concurrency of the three medians is called the centroid.