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
This challenge presented a very
unique problem, which may be of great interest to students desiring to
create balanced paper airplanes. The problems asked me to fold a piece of
paper down the center. I was then to create two folds, one from the top
right corner and the other from the top left corner, to any point on the
original fold. I was then to calculate where the paper should be folded in
a manner that would create a situation where the area of the triangle
created by the two diagonals and a point on the bisecting line would be
equal to twice the area of the rest of the paper.
Plans to
Solve/Investigate the Problem
I will try to work it out
folding paper first, and then move to GSP for pictorial representation.
Investigation/Exploration
of the Problem
I attempted to set the problem
up using a piece of paper. After many attempts, and getting close but still
not there, I switched to GSP.
I constructed a rectangle. I
then found the midpoints of the upper and lower line segments. When I
connected the two midpoints, WHALA!!! A line of symmetry was born. I then
selected a point on that line and created diagonals to each of the upper
corners. I calculated the area of the triangle and the two trapezoids.
Maneuvering the point on the line of symmetry, I was able to calculate
(almost) the point at which the area of the triangle was equal to each of
the trapezoids. I then created a perpendicular through the point on the
line of symmetry. Upon investigating the distances created on the two sides
where the perpendicular line intersected the rectangle, the relationship was
exactly 2 to 1.
See GSP Sketch below:
Extensions of the Problem
I further investigated that the
area of each triangle and the two rectangles are equal. You could further
investigate the relationships with the supplementary angles and different
bisectors.
Author & Contact
Jim Taylor
jtaylor1@rockdale.k12.ga.us
