**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.