Intermath | Workshop Support

 Write-up

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