Stephen J. Morgan

EMAT 6680

Assignment #4

Prompt:

Solution:

For this investigation we will be looking at centroids with respect to triangles. In this regard there are several facts that are true about centroids:

1.) The centroid will always lie inside the triangle (this is also true for any convex figure).

2.) If we think of a triangle XYZ, with points X, Y, Z, the centroid will always be C where C= 1/3 (X+Y+Z)

3.) A Centroid is 2/3 of the distance away from points X,Y, and Z on the lines that go through the respect midpoints of the triangles and form the centroid.

To start of this investigation we will keep these three facts in mind and look at the construction of various centroids of triangles. Click here to use GSP and make your own.

This script was originally created by Dr. Wilson and taken from this site, http://jwilson.coe.uga.edu/emt668/Asmt4/EMT668.Assign4.html. I decided, on my own, to make an acute,

right, and obtuse triangle and see if these properties hold. ***Note that there are hundreds, if not thousands, of different triangles we could come up with. We are doing this merely to

get our hands dirty and look at the situation deductively. It is important to realize even if we observe these properties for a million triangles, that alone is not enough to be assured of

the above claims. We have to prove them inductively, which we will do later through proofs. However for now, let's look at our three triangles.

Triangle #1: Right Triangle

Triangle #2: Acute Triangle

Triangle #3: Obtuse Triangle

We can easily see that the median lines have a 2:1 ratio for all of these triangles and from that can deduce C=1/3 (X+Y+Z). However, it is not

enough to make this claim merely from looking at examples we have to make an actual proof. Here's a simple proof showing why this is true.

Taking any arbitrary triangle: