Assignment 4:

Take me to your Centroid

by

Kristina Little

The centroid of a triangle is the point where all three medians intersect. A median is a line segment or ray that connects one of the vertices of the triangle to the midpoint of the opposite side. Below is a construction that should help you visualize a centroid.

An important idea is whether we have truly found the midpoint of the opposite sides. Algebraically, if we are given (or able to deduce) the equation of the lines for the three sides of the triangle, we have a formula we can use to find such a midpoint accurately. Geometrically, we know that the midpoint of each side will bisect that side into two equal parts. Using a compass and ruler we can easily recreat on paper the construction done through GSP.

Let's look at three specific tringles to understand how the centroid is influenced by and influences the triangle.

 Acute Triangle: Like the one in the introduction, the centroid is located inside the triangle and you can very clearly see each median from vertex to midpoint of each side. The medians are therefore concurrent inside the triangle and create the centroid. Right Triangle: Once more, the centroid in a point inside the triangle where all the medians are visible. Does the line segment BD look like anything in particular? Obtuse Triangle: Here again we see the centroid in located inside the triangle where all the medians are clearly visible from vertex to midpoint of the opposite side.

From these three specific examples of the types of triangles, one can conclude reasonably that the centroid will always be an interior point to a triangle.

An interesting trait of the centroid that should be mentioned is that any median through the centroid divides the triangle into two equal area triangles. In other words, line segment BD divides any given triangle (from above) into area of triangle ABD congruent to area of triangle CBD. A similar situation occurs for all other medians.

The centroid is also referred to as the balance point of the triangle, since if the triangle could be made of any material, then the centroid is where the triangle would balance.

Given the information about the medians and equal area, is the previous statement about the balance point of the centroid inferrable? Why or why not?