Assignment
#6
Exploring
Medians

Here
is a sketch of an equilateral triangle and it's medians.

On
this sketch, the medians are highlighted. Notice that if we constructed
a triangle with the three purple segments, the triangle would
also be equilateral.
Observations:

1.) The
segments must be the same size because of SAS congruence.

For instance,
we know that AB = AC and AM1 = CM2 and all angles of the original
triangle are 60 degrees by definition of equilateral triangles.
Since we know we have congruence, we know that the corresponding
parts of congruent triangles are congruent.

Let's
look at an isosceles triangle:

Notice
that the medians are the yellow segments. Let's check to see if
the medians will create another isosceles triangle.

So
again, the medians create another isosceles triangle.

Next,
we will explore to see if the same relationship exists for right
isosceles triangles.

Now
we will check the medians.
The
pattern is the same for right isosceles triangles.

My
last investigation will be whether a right triangle will always
generate a right triangle of medians.
Let's
look at a right triangle.
Now
let's see if the medians will create a right triangle as well.

As
you can see, the medians do not create a right triangle.

From
this investigation we have learned that equilateral and isoceles
triangles create medians that have similar qualities. However,
right triangles will not generate medians that make right triangles.
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