Medians Do What?

by

Priscilla Alexander

This write-up is for geometry or trigonometry students learning about the median triangles of right triangles and isosceles triangles.

The median of a triangle is formed by taking the midpoint of a side of a triangle and connecting it to the vertex that is opposite of it. See picture below.

A median triangle is formed by taking the medians of all sides and then translating the sides to form a triangle see picture below.

Will an isosceles triangle generate isosceles triangle of medians. In other words, will the median triangle of an isosceles triangle be an isoscles triangle. The answer is yes. The median triangle will always form an isosceles triangle. No matter how the isosceles triangles angles are changed the median triangle will be an isoscele triangles. Click on the below link to test it out.

GSP Isosceles Triangle and Medians

Why is the median triangle an isosceles triangle?

The first reason is two sides of the median triangle are equilateral. Equilateral means they are the same length. The second reason the base angles are congruent. Congruent means they are the same.

Try constructing your own median triangle in GSP below

Will a right triangle always generate a right triangle of medians? No a right triangle will not generate at least one right median triangle. See GSP sketch below.

GSP Right Triangle and Medians

Why does this happen?

When taking the median of triangle WTU the median triangle formed is ZTA. When using Pythagorean's Theorem to try and prove that the median triangle is a right triangle it does not work because the two sides added together and then squared does not give the hypothenuse.Therefore, the median triangle cannot be a right triangle. It is not an isosceles triangle because two of the sides are not congruent. The median triangle cannot be equilateral because none of the sides are congruent. The only conclusion that can be made about the median triangle of a right triangle is that it is a scalene triangle. A scalene triangle is a triangle that has no congruent sides. No matter how the lengths of the right triangle are changed the median triangle will be a scalene triangle.

GSP Right Triangle and Medians

The only special case is when the right triangle is an right isosceles triangle. Then the median triangle that is formed will be an isosceles right triangle.

Try it out on the following sketch.

In conclusion, medians of right triangles and isosceles triangles form different types of triangles from the original triangle. A right trianglewill form a median triangle that is a scalene triangle. The isosceles triangle will always form isosceles median triangles.