The Medians of a Triangle
by Mark Cowart (EMT668/Summer 1998)
A median of a triangle is a segment from the vertex of the triangle to the midpoint of the opposite side. Every triangle has three medians. Click here to see an illustration.
Students may find interest in examining all three medians in a triangle at the same time. They will intersect at one point. This intersection point or point of concurrency is called the centroid.
Explore the following:
1) Open a GSP sketch and construct a triangle of any type or size (label the vertices A,B,and C). Next find the midpoint of segments AB, BC, and AC (label these G, H, and I, respectively). Do the medians intersect at one point? (label the centroid point X - you may need to highlight two of the medians and construct 'point of intersection').
2) Find the measure (either inches or centimeters) of segment AX and XH. Do you notice anything special when comparing these two lengths? Try dragging vertex A of the triangle until segment AX is 2 in. or cm. What do you notice about XH? Try finding the lengths of the other median portions and examining/comparing their lengths. Do your findings change when the shape of the triangle is different (acute, right, or obtuse)?
3) A pattern of comparison between the median parts (from vertex to centroid and from centroid to opposite midpoint) should begin to become clear. Click here to see a comparison or confirm your conclusions. There is another conclusion that needs to be drawn which describes the relationship between the centroid and its position along each median. Compare some of the lengths from vertex to centroid with the entire median length it lies on. Complete this conclusion: The point of concurreny of the medians is at a point that is________________________________________________.
4) Can you prove your theory about the centroid? Try elaborating with a proof in T-form or less formally.
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