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.