**Assignment #6**

**Investigations
of a Triangle formed by the Median**

**Lengths
of Another Triangle**

**by**

**Kimberly
Burrell**

**Given triangle ABC with medians AD, BE,
and CF, we can construct a triangle in which the three sides have
the lengths AD, BE, and CF. At this time we want to investigate
the relationship between the original triangle and the new one.**

**After constructing these two triangles,
we find the perimeter and area of each. We can then calculate the
ratio of the triangle formed by the medians and the original
triangle with respect to both perimeter and area.**

**Notice that the ratio of the perimeters
is not included, however we see the ratio of the perimeters
squared. Comparing this particular ratio to the ratio of areas,
we are able to find an equal ratio. Will this relationship be
consistant with any triangle? ****Click here****
to try.**

**You will notice that as you manipulate
the original triangle, the ratio of the area remains constant at
0.75. Can you find a case where the ratio of the perimeters
squared is not 0.75? The question to explore and investigate now
becomes when is the ratio of the perimeters squared not 0.75?**