**Assignment 6, Problem #1**

**Construct a triangle and
its medians. Construct a second triangle with the three sides
having the lengths of the three medians from your first triangle.
Find some relationship between the two triangles. (E.g., are they
congruent? similar? have same area? same perimeter? ratio of areas?
ratio or perimeters?) Prove whatever you find.**

**Go to Text
for Figure 1**

**Go to Text for Figure 2**

**Go to Text
for Figure 3**

**In Figure 1, triangle ABC was constructed using three arbitrary
points. Once the triangles were constructed, the midpoints or
medians were also constructed. I decided to connect the three
medians to make a second triangle to see if the this new triangle
was going to have the lengths
of the three medians from my first triangle. They did!! I do not know why, but I never really
realized this...maybe I need to have more investigative studies
for my students so that they may have beautiful discoveries such
as this simple one!**

**In Figure 2, the equilateral triangle ABC is given with its correlating
sides. If you notice, the sides of triangle FED are the lengths
of the medians of triangle ABC. Also, triangle FED is similar,
not congruent, to triangle ABC. It is similar to triangle because
the segments in ABC are twice the length of the segments in triangle
FED.**

**In Figure 3, notice that the perimeter of triangle ABC is twice
that of triangle FED. Also look at the areas of both triangles.
What do you notice here? What can you say about the ratios of
the larger triangle to the smaller triangle? Could you create
another triangle, say an obtuse triangle and the outcomes remain
the same? Can you make a conjecture about your findings?**