Philippa M. Rhodes

## Write-up 6

Construct a triangle and its medians. Construct a second triangle with the three sides having the lenths of the three medians from your first triangle. Find some relationship between the two triangles.

The median of a triangle is the line from a vertex to the midpoint of the opposite side. We will first construct triangle ABC and its medians.

Next, use the lengths of the three medians to construct the sides of a second triangle.

Remark It will suffice to examine only one triangle drawn using the lengths of the medians of triangle ABC since all such triangles are congruent. We know this because SSS = SSS or because the ratio of the areas is one.

We can now begin to compare the two triangles. They may appear to be congruent or at least similar.

Here are some measurements of the two triangles.

The chart is set up to compare what had appeared to be the corresonding sides and angles. Now we can easily see that the two triangles are not congruent nor are they similar.

One way that we know that the triangles are not congruent is by noticing that none of the measurements of the sides nor of the angles are equal. Another way is to compare the areas. As stated earlier, the ratio of the areas of two congruent triangles is one. Well, Area GJL / Area BAC = 0.75 (close, but not congruent).

Using GSP, we are able to change the size and shape of the original triangle which changes the size and shape of the triangle of medians. By doing so, we notice that the ratio of the areas of the two triangles stays 0.75 while the ratio of the perimeters varies slightly.