from Assignment 4, #14

The Centroid

by

Megan Dickerson

High school students often have a skewed understanding of proofs. They enter Geometry class normally thinking that if it looks true, then it must be true. However, that is a very bad assumption. Proving something takes away all assumption and replaces it with truth. In this write-up I will prove that the 3 medians of a triangle concur and that this point concurrence is 2/3 of the distance up from the vertex to the opposite side. I will use GSP to visually show my steps, and I will explain how this can be used to give students a better sense of proof.

First of all, when students look at the triangle below it LOOKS like the three medians concur. They do in fact concur, but it is not enough to just say "because the picture shows that they do..." We must justify WHY and give PROOF!

To do this proof I am going to use similar triangles and parallel lines.

I will start with my Triangle ABC and just 2 of the medians, AE and CD. AE and CD are not parallel, so the must meet at a point, P.

 Connect DE. This segment is parallel to the base AC since D and E are both midpoints . Since DE and AC are parallel, angles marked are congruent. Plus angle DPE and angle APC are vertical angles, thus congruent.

So, triangles DPE and CPA are similar since they have congruent angles.

Then, since triangles DBE and ABC are also similar and DB is half of AB and BE is half of BC, then DE is half of AC.

Since, DE and AC are in a 2:1 ratio, all sides of triangles DPE and CPA are in a 2:1 ratio.

So, the point of concurrence of these 2 medians, CD and AE is at P, and P is 2/3 of the way up from the vertex on both of these medians.

Now I need to prove that P is in fact the centroid.

To do this I will now look at only medians AE and BF.

Similar to before, these 2 medians must intersect at a point since they are not parallel. Call this point Q. Connect EF. EF is parallel to AB since E and F are both midpoints. So, in similar argument as the other 2 medians, triangles EQF and AQB are similar with congruent angles. EF is half of AB, so the sides of these 2 triangles are also in ratio 2:1.

Thus, Q is the point of concurrence of these 2 medians and is 2/3 of the way up from the vertices on the medians.

However, P was the point 2/3 of the way up from AE, so P must equal Q.

This point is the centroid.

In the classroom I would use GSP to visually show, like I did, each step.

I would also use it to show the ratios below:

The fact that all 3 ratios are 2/1, shows that the centroid is 2/3 of the way up from the vertex.

I would also use GSP to move the shape of the triangle and show that no matter the shape of the triangle, the ratios still hold.

The word "proof" often scares students because they associate it with a lot of difficult work. GSP makes the process a little more interesting and interactive. The students are able to clearly see how the proof works (my drawings on the board are not as pretty or accurate).