** **

**Triangle medians**

**By **

**Princess Browne **

We will construct a triangle
and its medians. Then, we will construct a second triangle with the three sides
having the lengths of the three medians from the first triangle. Finally, we
will find the relationships between the two triangles. Some examples of the
relationship are as followed: are they congruent? Similar? have same area? same
perimeter? ratio of areas? ratio or perimeters?

I will start by constructing
triangle number 1 and its medians. Then, I will construct triangle number 2
using the medians of triangle number 1.

Given DABC, let the points D, E, and F be the midpoints of
segments AB, AC and BC. I know
that segment AD È DB, AE ÈEC, and BF È FC by the definition of midpoints. If we connect the three midpoints
to form triangle number 2, we will end up with a median triangle.{ERROR. The medians are segments from a vertex to the midpoint of an opposite side.} By the
definition of a mid segment of a triangle I know that the mid segment of DABC is DE. I also know that segment DE is 1/2BC, DF is
1/2AC, and EF is 1/2AB. With this information I know that DABC and DDEF have a ratio that is 2:1. A ratio of the perimeters
that is 2:1.

Below are the lengths of the
different segments

However, the ratio of the
area of these triangles is different from the ratio of the perimeter because
there are four of the smaller triangles within the original triangle. Therefore
we can conclude that the ratio of the area of a triangle to itÕs median is 4:1.

since the ratios of the sides
are not always the same the triangle are not similar, but the perimeters and
areas of the two triangle will always have the same ratio. The relationship
between the two triangles is when the original triangle is an isosceles triangle
the median triangle will also be an isosceles triangle. This is also true for
right triangles, obtuse triangles and equilateral triangles