Exploring Triangles and its Medians

by

Rita Meyers

This write-up will explore a triangle and a triangle
whose sides are equal to the lengths of the original triangle's
medians.
We will first start by constructing
a triangle from the medians of the original triangle. I
will call this triangle the medial triangle.

So, given this original triangle ABC and its medians.
We construct the medial
triangle DEF.

Once we have both triangles let's look at their relationship to
one another.

First we will measure the medians of the original triangle and
verify that they are in fact equal.
As we see the lengths are in fact equal
Now, let's look at the area of
both the original triangle and the medial triangle.
Right off the bat we see no relation
between the two...they are not equal.

Let's look at the ratio of the area of triangle DEF to triangle
ABC
It appears that the area of triangle
DEF is 3/4 the area of triangle ABC.
Is this always the case? Let's look!

Exploration #1
Moving pt B of our original triangle
along a segment, we notice that it changes the size of not only
the original triangle but also the medial triangle.
We can also see the changes in the
calculation of the areas of both triangle.
More importantly, we notice that the
ratio between the two areas of the triangles does not change.
Let's look at it another way.
This time we will move pt A of our original triangle along the
circumference of a circle and look for changes in the ratio of
the areas.
Exploration
#2
No matter if the original triangle
is acute or obtuse, the ratio of the area of the original triangle
to the area of the triangle made up of its medians does not change.
The area of the medial triangle is always 3/4 of the original
triangle.

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