Bisecting the Area of a Triangle

By

Jeffrey R. Frye

Problem 1:  For any triangle, construct a segment parallel to a base of the triangle that divides the triangle into two equal areas.

When a segment is constructed parallel to the base of the original triangle, two similar triangles are made.  They are proved to be similar by the AA similarity postulate.  Since the triangles are similar some proportionality statements can be made.  From the triangle below, we can state that.

This statement can be written in terms of m so that.  The problem is to show that the area of the upper triangle is half the area of the original triangle.  This means that the relationship between the areas could be written as.  Simplifying this equation will result in

.  By using substitution, we can derive that.  Thus, when the top triangle is half the area of the original triangle, the length of the new parallel base will be of the relationship     The ratio of the areas of the upper triangle and the original triangle is, and the area of the quadrilateral is equal to the area of the new triangle.  The GSP sketch will show the construction of the parallel base of the triangle constructed to this base length relationship.  The sketch confirms the length ratio and area relationship by measurement.  Click here to view the sketch and to make your investigation.

Problem 2a:  If the parallel segment that divides the triangle into two equal areas is drawn for each base, a smaller triangle is formed. What is the ratio of the area of the small triangle to the original?

From the figure, the smaller triangle is constructed as shown.  The ratio of the area to the larger triangle is .0147.

Problem 2b:  The segments parallel to the bases divide the original triangles into two equal areas.  What is the ratio of the area of the shaded triangle to the area of the original triangle in the figure below?  Since the constructed triangle is part of each half the ratio to each is the same, .34.  The smaller triangle area to the area of the original triangle area is .17.

Problem 2c:  Prove that the measures of the three shaded areas in each of the figures below are the same. In each figure what is the ratio of the area of one of the regions to the area of the original triangle?

There are some interesting relationships that result from drawing the lines parallel to each base.  When the small triangle area is added to the adjacent quadrilaterals to form triangles, the area in each of these three triangles is equal to the area in each of the other three quadrilaterals.  The small triangle is located where the centroid of the large triangle is found.  These images are shown below.  Click here to open GSP to explore these relationships.