# Dividing a Triangle Into Two Equal Areas

### and

# Exploring Area Ratios

### by

## Alison Hays

## In this essay, I will explore the line parallel to a base
of a triangle that divides the triangle into two equal areas,
and I will explore the areas of the sections of the triangle that
are created when all three such lines are constructed.

For a statement of this problem, please click here.

There are several parts to this problem, and I will answer
each part in turn.

### Part 1: Construct a segment parallel to a base of a triangle
that divides the triangle into two equal areas.

For a description of the construction,
click here.

To view a GSP sketch of the construction,
click here.

For a GSP 3 script of the construction,
click here.

### Part 2: 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?

### Part 3: What is the ratio of the area of the shaded triangle
to the area of the original triangle in the figure below? Here
again the segments parallel to the bases divide the original triangles
into two equal areas.

### Part 4: 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?

### Extensions:

Are there any other relationships among areas of the triangle?

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