InterMath | Workshop Support | Write Up Template
March 2, 2004
Title
Paper Folding
Problem Statement
Take a rectangular sheet of paper. Fold it in half to make a crease down the center of the
sheet from top to bottom. Then, select a point on the sheet and make a crease from the
upper right corner to the point; now make a crease from the upper left corner to the point.
How would the point be selected so that the triangle formed by the top of the sheet and the
two slant creases has the same area as each of the lateral trapezoids?
First, get some sheets of paper and do some folding to get a feel for the problem. It's
possible to make a fold across any two points, and a point is indicated where two creases
cross or where a crease intersects an edge of the paper.
Folds can be used to bisect a line segment. For example, the bottom of the page is a line
segment. We can match the corners of the page together to form a crease that is the
perpendicular bisector. Proof?
Second, folding to trisect a line segment (e.g. folding the paper into thirds) is probably a
guessing game. If you claim it is a "folding" construction you should have a proof that the
fold trisects the segment (exactly, not approximately.)
Third, of course you will want to switch to a line drawing representation for analysis and
proof at some point. Use similarity concepts to show an exact folding construction for the
desired configuration.
Problem setup
Folding a sheet of rectangular paper to determine the point that would make the area of the
triangle and the area of the lateral trapezoids congruent may not be accurate, but it is a great way
to begin. It gives a better understanding about what the problem is discussing. Putting the
problem into Geometer’s Sketchpad will make it easier to see that the areas of each polygon are
congruent. It is very important to recognize that the vertical line of symmetry (runs through the
midpoints) must maintain a point that exemplifies a relationship between in some way.
Plans to Solve/Investigate the Problem
I plan to use Geometer’s Sketchpad to construct the rectangle. Using the interior of the
rectangle, I will construct a vertical line that will help identify the triangle and two lateral
trapezoids inside the rectangle. I plan to use the point on the vertical line to investigate the
similarity between the triangle and trapezoids. The point is extremely critical to solving the
problem because it will develop several outcomes to form a proof of the problem.
Investigation/Exploration of the Problem
Exploring the problem became more apparent after the construction of the rectangle. The
rectangle was formed by the intersection of perpendicular lines. From the midpoints of opposite
parallel sides of the rectangle, I formed a vertical line. Placing a random point on the line, led
me to move it up and down the vertical line to see if there was any change. Next, I constructed
the legs of the triangle which also formed a side of each trapezoid. I finally got the point at a
place that would make the area of both trapezoids and the triangle equal. Measuring the area of
each polygon was easily performed in Geometer’s Sketchpad. Remember that area of a triangle
= ½ base x height and area of a trapezoid = height ( base 1 + base 2) / 2. Now, the point is
exactly where it needs to be along the vertical line. Based on the measurements of the legs of the
triangle, I found that the triangle is isosceles. More importantly, I found that the shared side of
the trapezoid (segment NJ) makes up 1/3 of the total length of the line segment (PN).
m PN = 4.82 cm
m PN
NJ
= 3.03
This shows me that the point on the vertical line would be placed a third
of the way from the bottom of the page. To prove that the segment must be divided into thirds, I
also found that the same segment NJ is ½ of segment PJ. This is performed in a ratio at the
bottom of the sketch.
Measurements of legs of
triangle:
JL = 4.08 cm
JK = 4.08 cm
K
P
L
Area of Triangle:
Area
KL = 5.00 cm
JKL = 8.07 cm2
Area of
Trapezoids:
J
Area KMNJ = 8.00 cm2
Area LJNO = 8.00 cm2
M
N
O
NJ = 1.59 cm
JP = 3.23 cm
NJ
JP
= 0.49
Extensions of the Problem
Going back and folding the paper to prove the problem would be a possible extension. Folding
the paper in half vertically, gave me the line in which my point would be placed. Turning the
paper in the opposite direction (vertical line now runs horizontal) will make it easier to fold the
paper into thirds. The most accurate way I have found to fold the paper into thirds would be to
fold it like a cylinder and flatten it out to create three equal sections. Turning the paper upright
will show three equidistant points along the vertical line. The third point (closest to the bottom
of the page) would be the shared point of the triangle and the trapezoids that lies along the
vertical line. Using a ruler to measure the distances needed for the area of each polygon would
be a way to solve the extension. I think it is a great idea to see it both ways. Remember that area
of a triangle = ½ base x height and area of a trapezoid = height ( base 1 + base 2) / 2.
Author & Contact
Nicole Vater
nvater@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2