Assignment #6 ~ It's More Fun on the Medians!
Triangle Construction Given the Medians
By Kevin Mylod

If we are given three arbitrary segments AB, CD, and EF (shown below), which are the medians of an unknown triangle, can we construct the triangle? AB, CD, and EF have arbitrary lengths of j, k, and m respectively.

First, let's construct a triangle made up of the three medians AB, CD, and EF.

We know that these medians are concurrent within the unknown triangle and that they intersect a point that is 1/3 the distance of each median. From another problem with in this assignment we learned that we can construct two of the three intersecting medians, say segments m and k, by constructing a parallel line to k through a point, P, on m that is 1/3 the distance and marking off k'.

The definition of a median of a triangle is a segment where its endpoints are a vertex of the triangle and the midpoint of the opposite side. As a result of constructing the intersection of two of the medians, we also have constructed two of the vertices of the unknown triangle. The next figure constructs that segment as well as its midpoint, X.

By definition, we know that the third median, j', will go through P and have X as one of its endpoints. For the same reasons k is parallel to k', so will j to j'. We can construct the other endpoint, R, of j' by a simple center point-radius construction.

We now have our unknown triangle RST. But do these medians actually go through the midpoints of triangle RST? One way to find out is by changing the lengths of the medians to see if the midpoints stay in tact. Click here to explore this in Geometers Sketchpad. To investigate the construction of a triangle given the medians, please take a moment to look at the script.

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