Explorations with Geometer’s Sketchpad

Assignment 6

Problem #3

By Erin Cain

In this problem we are asked the following:

Given three line segments as the three medians of the triangle. Construct the triangle.

If you would like to try this for yourself, click HERE to open a GSP file that has three given medians for you to try.

In order to solve this problem I did the following:

First I began by constructing a triangle that has the three given medians as its sides.

Next, I constructed the centroid of this triangle which is the intersection of the medians of an original triangle.

The third step I took was to construct a circle that has a center which is the centroid and has a radius that is the distance from the centroid to one of the vertices (you can choose any of three distances).

Now we need to extend the median whose vertex you used to determine the radius of a circle into a line. The intersection of the median line and the circle (not the vertex) will be the midpoint of one of the sides of the triangle you are constructing. The vertex of the purple and light blue side will also be a point on the triangle we are constructing. We can now use this to create one of the sides of your triangle by making another circle with the center at the midpoint (M1) and going through the vertex of the triangle. The neon green line segment in the picture below represents one of the sides of your new triangle.

Since we have one side of our new triangle, we also have two of the vertices (V1 and V2) of the triangle. Now if we can form another side of the triangle, we will have all three vertices!!!

Therefore, in order to form another side of the triangle we need to look at the median through the vertex of the purple and light blue side of our median triangle. The length of our median is actually 3/4 of the size of the side we want. Thus, we need to construct another circle that has a center at the midpoint of the pink side and that passes through the centroid of our median triangle. If we then extend the median, the intersection of the median and the circle is the third vertex (V3) of the triangle we are building.

In order to find our triangle, we simply need to connect the three vertices together by constructing line segments from V3 to V1 and V3 to V2. Our triangle is the neon green triangle below:

We can check that this triangle has the medians given by measurement:

RETURN