Assignment 4: Investigating Triangles

by

Melissa Silverman

The centroid of a triangle is the point of intersection for the three lines from each vertex to the midpoint of the side opposite the vertex.

How is the centroid's position affected by the shape of the triangle? Using the GSP trace tool, one can change the triangle and track the movement of the centroid. In this first construction, we are experimenting with moving only one vertex, Point A, along a line. When Point A is moved along the line, the centroid's path is also a straight line. Its path is parallel to the line that Point A is being moved on. Notice that the height of the triangle changes as Point A moves, but the centroid continues to move along a straight path.

When one point of the triangle is moved along a line, then the centroid follows along a straight line. What happens if more than one point of the triangle is moved? In the following construction, two points of the triangle are moved along a line.

The centroid moves along a rectangular path. As points A and B move along the gray lines, the centroid traces a rectangle. So when two points are moved along perpendicular lines, the centroid follows a rectangular path. Similar to before, the cetroid moves along an orderly path, as opposed to jumping all over the construction.

What happens if all three points are moved along a straight line?

The path of the centroid is a parallelogram. Again, as points A, B, and C move back and forth along their lines, the centroid follows a predictable path.

These past few examples only explore what happens when the points travel along perpendicular lines. What happens if the lines are constructed at an angle. Does the centroid still follow a particular path?

The centroid follows along a lattice in the shape of a parallelogram. The path is not as orderly as when the lines are at right angles, but the path of the centroid does trace a particular design composed of straight lines and right angles.

What if the vertexes of the triangle were moved around a circular path? Moving two of the three points around the circle, while tracing the centroid, results in a path that looks like a flower.

This exercise fostered a lot of curiousity and excitement. As the constructions got more complex and the centroid paths more interesting, peers began to peek over and give their input. We had fun inventing new ways of moving the triangle and predicting the effect that that movement would have on the centroid's path. This exercise is visually enticing, so it sparks interest and lots of communication between students.
I also like the fact that the constructions required for this exercise are easy, but the results are impressive. Students with little practice on GSP could complete this exercise. It is a great way to break into GSP's capabilities, especially with the animation tool.

I was also very impressed by the power of the trace tool in the investigation. It is easy to make a triangle and use a GSP tool to find the centroid. It is also easy to just move around the vertices of the triangle and watch the centroid wiggle. But when the centroid is traced and its path is drawn, it really brings up some neat points. Prior to tracing the centroid, I did not think much about where it was going as the triangle around it changed. My hypothesis was that the centroid would not follow an orderly path. But if the vertices of the triangle are moved on a path, the cetroid draws a very clear path. Even after changing the characteristics of the animations (speed, direction, etc.), the centroid paths were controlled.

The only thing that bothers me about this discovery is that I am not sure how much mathematics this is supporting. It helps to understand the relationship between the centroid and the triangle, but I am afraid I cannot see too many educational applications of this discovery.
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