Assignment #6

Constructing a Triangle Given Its Medians

by

Angela Wall

Given three line segements as the medians of a triangle, construct the original triangle.

Let segments AB, BC, and AC be the medians of a triangle. With the three given line segments as the medians of a triangle, use theses segments to create another triangle.

When all three medians of a triangle are found, they intersect at the centroid. One property of the centroid is that it is 2/3rds along each median in the triangle. This fact will be used several times to construct the triangle with the given medians.

First, choose two of the given medians, and trisect the segments. Below, segment AB and BC are trisected. Trisecting each segment breaks it into three congruent sections.

Choosing segment AB, construct a segment congruent to BC through one of the trisecting points on AB. Call this segment DE. DE will need to intersect AB so that the point of intersection is one of the trisecting points on DE. To do this, take 2/3 the length of BC and draw a segment. Using this segment as a radius and one of the points on AB as a center, construct the circle. Do the same thing for the remaining 1/3 of BC by constructing another circle. Through the center of the two circles, construct a line parallel to BC. Mark the appropriate points of intersection with the parallel line and the circles and construct a line segment. This segment is DE. Notice that since we how have two intersecting medians, the point of intersection is the median. So, point F is the centroid of the triangle we are trying to construct.

Constructing segment BE will be one side of the triangle we are trying to construct. Notice that points B and E are two thirds the length from point F on their corresponding median.

Next, we know that the median that has not been used has to connect segment BE to its opposite vertex. By definition of the median, we know that the remaining median will intersect BE at its midpoint. So, to construct median AC, first find the midpoint of side BE. Using the midpoint as center, construct a circle with a radius the length of AC. We know that this median will have to go through the centroid, point F. Thus, construct a line through the center of the circle, through point F. Mark the point of intersection with the circle and line. Note we are interested in the point of intersection that is closest to the centroid, point F. Call this point of intersection G. Thus, the distance between the midpoint of BE and G is the length of the final median.

Finally, we see that we can construct our original triangle by connecting points B, E, and G.

Checking the construction, finding the centroid of triangle BEG results in the same point as point F.