Assignment 6

This activity explores the properties of a triangle whose sides are the medians of some original triangle

For this exploration we will construct a triangle and its medians. We will then a second triangle with the three sides having the lengths of the three medians from your first triangle. The focus of this exploration is to investigate the relationship between these two triangles.

First construct a triangle and its medians.

Next, construct a second triangle with the three sides having the lengths of the three medians from your first triangle.

The goal of this exploration is to find some relationship between the two triangles.

Click here for an interactive java sketchpad sketch of these two triangle.  Use this sketch to investigate the relationship of these two triangles.  What do you notice about the perimeters?  About the areas?  What about the ratio of perimeters?  The ratio of areas?

This section will investigate the relationship shown below.  That is, that the ratio of the area of the median triangle to the area of the original triangle is .75 or 3/4.

Another way of representing this concept is the picture below.  The proof that follow uses this next figure as a reference.

Proof:
Suppose the area of triangle ABC is 24.  (This number is chosen because it is divisible by 2, 3, 4, and 6.)  Given that BE is a median (BE divides ABC into two equal pieces),  the area of triangle ABE is 12.  We know that GE = BE/3, because G is the centroid of the triangle.  Then the area of AGE is 12/3 or 4.  FK is parallel to BE, by construction, and F is the midpoint of BC, therefore, H is the midpoint of EC.  Then AE/AH = 2/3.  It follows that the area of AGE / the area of AFH = (2/3) (2/3) or 4/9.  Since the area of AGE = 4, the area of AFH = 9.  Since H is the midpoint of FK, then the area of AFH is one half the area of AFK.  So the area of AFK is 18 which is 3/4 the area we started with.