By: Shelly B. Kaufmann
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.
Once you are comfortable with the paper folding, make a GSP sketch!
Remember, your sketch will not necessarily have the same measurements as mine!
Draw a rectangle to represent the sheet of paper you just worked with. Label the vertices AEFC. Find the midpoints of AC and EF then draw a perpendicular line. You might want to make this a dashed line. Next, locate the midpoints of ED and DF. Use these points and each of the corner vertices to construct segments to the newly established midpoints. Where the segments cross your perpendicular line, should be labeled B. This is the point you need to have 3 equal area parts!
You can clean up your drawing now that you have B. Hide lines that you constructed last then join only the segments needed – AB and CB. You could also hide the midpoints of ED and DF.
The final step once you have your drawing is to find the area of each of the figures within the rectangle. Highlight each of the figures in the rectangle to find the sum. The sum of the areas of the 3 figures should be the total area of the rectangle!
Are your measures different from mine? J
What did you learn about point B?