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.

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