“Dividing a Sheet of Paper into Three Equal Thirds with one Equilateral Triangle”

by

Christopher R. Whitworth

You have been given
a blank sheet of paper (8.50 cm x 11.01 cm), and have been instructed to divide
the sheet of paper into thirds (in area) while one of the thirds represents an
equilateral triangle. The blank
sheet of paper is shown below.
(Note: Some values may not
be completely accurate and may differ by no more than 1/50^{th} of a
centimeter).

The next step is to
create a midpoint between each of the two line segments, FH and GI. Then connect the two midpoints with a
dotted line that divides the sheet of paper in half.

Then, you should
create two lines that are 8.50 cm in length each that connects point G to the
red-dotted line and I to the red-dotted line. The reason you are selecting 8.50 cm as your value is
because you want *at least* one equilateral triangle
(three equal sides and angles).
The bottom of the sheet of paper is already 8.50 cm, so if you make the
lines 8.50 cm and have them intersect at one point, then you have created an
equilateral triangle.

Next, you will
want to hide the section of the dotted line that lies just below point J.

The next step
will be to fill in each of the three sections a different color so that you can
distinguish between the three sections.

Then, you will calculate the area of each section by using either Geometers Sketchpad 4.0 or a ruler and the formula for area (A = L x W).

Notice that
the area of each section is equal to 31.19 cm. This means that the area of each triangle is equal. I have shown you how to divide a sheet
of paper into three sections with equal areas including one equilateral
triangle. Below is the pattern on
how to fold the paper to determine the three equal sections.

You have just
folded the paper in half. Now,
unfold the paper and fold the right corner to the fold in the center. Then, draw a point on the paper where
the corner of the paper touched the fold.
Do not fold the corner back to the original position. Next, you will want to repeat this
process with the left corner and the point on the fold in the center.

Finally, you
will notice a shape that looks similar to the one below. Once you have constructed this figure,
unfold the paper to its original position and measure the area of each
section. You will notice that the
area in each figure will be equal to one other.

Once you have
determined that each figure’s area is equal to one another, then you can
prove that you can divide a sheet of paper that is 8.50 cm x 11.00 cm into
three sections with equal areas while including an equilateral triangle as one
of the sections.

For questions or comments, contact me
at: