Assignment Four
Problem #4

The purpose of this problem is to investigate the CENTROID of a triangle and explore its location for various shapes of triangles. The CENTROID of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.

In triangle ABC, the midpoint of side AB is E, the midpoint of side BC is F, and the midpoint of AC is D. The verify the midpoints, we can measure the segments from the vertices to the midpoint.

The medians are drawn from the vertices to the midpoints opposite. Median AF (in blue), median BD (in green), and median CE (in red) intersect at point G. This point is the CENTROID of triangle ABC.

There are several interesting mathematical finds we can discover about the centroid, and we will examine at two of them. First, we will measure the segments between the vertices and the centroid and the segments between the midpoints and the centroid. Then, we will compare these segments as ratios.

VERTEX TO CENTROID---------MIDPOINT TO CENTROID

RATIOS

(Note that 5/2 does not equal two as the first ratio implies. The preferences of the Geometer's Sketchpad is set on units which are rounded to the nearest whole number. This makes the measures less precise. But when we use the segments themselves, the measures are more accurate.)

Can we conclude that the ratio of the segments between the vertices and centroid to the segments between the midpoints is always two? Let us examine what happens when the dimensions of the triangle change.

The ratio of the segments between the vertices and the centroid to the segments between the midpoints and the centroid remains two to one. It now can be concluded that the location of the centroid with respect to the ratio of the interior segments remains the same.

Now, we will examine the areas of the smaller triangles inside the centroid triangle which we have given different colors.

Note the areas are the same. Will this happen if the dimensions of the triangle are changed?

The areas of the interior triangles of the centroid triangle remain the same.

Click here for further exploration.

For the remainder of this lab, I would ask my students to explore ideas of the centroid, including comparison of the perimeters of the interior triangles to determine if there is a relationship, test congruency of the interior triangles, copy and paste the interior triangles to translate, rotate, or slide into another geometric figure, check the relationships of the ratios of other segments or angles, etc. Evaluation would be based on thinking processes and creativity as well as accuracy and use of the sketchpad.

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