Math Origami

July 6, 2015

Samantha is given a piece of square Origami paper, and given the following set of instructions to divide the paper into three congruent pieces without a ruler.

Step 1:  Make a small crease to mark the paper halfway up the side by folding the paper in half. (Fig. 1)

Step 2:  Create two diagonal folds,

  1. the first across the square from the top left corner to the bottom right corner and
  2. the second from the bottom left corner to the halfway mark from Step 1. (Fig. 2)

Step 3: Fold the top of the paper down to where the two diagonals intersect and create a horizontal fold across the square. (Fig. 3)

Step 4: Fold the bottom of the paper up to meet the horizontal crease and create another crease straight across. (Fig. 3)

The two horizontal creases, Samantha is told, divide the paper into thirds. 

Samantha wants to prove this is true, so she imagines the square is on a coordinate plane. She labels the vertices with the coordinates A(0, 0), B(0, 6), C(6, 6) and D(6, 0), as shown (Fig. 4).  Based on these coordinates, what are the equations of the two diagonal folds? Express your answer as an equation in slope-intercept form (y = mx + b).


Samantha then labels the intersection of the two diagonal folds point E and determines its coordinates. What are the coordinates of E? Express your answer as an ordered pair.


Samantha uses this information to find the distances between the top of the paper and the upper horizontal crease, the distance between the upper horizontal crease and the lower horizontal crease, and the distance between the lower horizontal crease and the bottom of the paper. Assuming, these distances are equal, what is the distance Samantha calculates?