# 1. Balancing a Triangle

Each team of students is handed a cardboard triangle and asked to find the point at which the triangle will balance on, say, a ChapStick® container (or any small cylinder, such as a stack of dimes).

This is an ideal activity for introducing the idea of a MathLab, since it is very easy to state and to fit into the guidelines of the three requirements, Q, R, and V.

Recall that the centroid, or center of mass, of a triangle is the intersection of a triangle’s medians. By definition, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. By theorem (this is generally proved in a single variable integral calculus course), the three medians will intersect at a common point, called the centroid of the triangle, and this point is the center of mass of the triangle. Thus, the students merely have to draw the medians to discover that they do indeed intersect at a common point, which is the desired centroid.

Thus, all three requirements are met:

Q. The question is simply that of “finding the triangle’s center of mass.”

R. The matter is easily resolved by the simple mathematics of finding the midpoint of each side, connecting each to its opposite vertex via straight line, and observing the emergence of the centroid.

V. That this point is indeed the center of mass of the triangle is easily verified by actually balancing it on the designated cylinder.