Assignment #4

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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