Craft in Geometry

1. A piece of paper is folded in half. A second fold is made at an angle φ(0<φ<90 span="">◦)to the first,and a cut is made as shown below.

When the piece of paper is unfolded, the resulting hole is a polygon. Let O be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at O, and also that ∠XOY = 144◦, where X and Y are the vertices of the hole adjacent to O. Find the value(s) of φ (in degrees).

Solution:

Try actually folding a piece of paper. We see that the cut out area is a kite. The fold was made on AC, and then BE and DE. Since DC was folded onto DA, we have∠ADE = ∠CDE. Either A or C is the center of the circle. If it’s A, then ∠BAD = 144◦, so ∠CAD = 72◦. Using CA = DA, we see that ∠ACD = ∠ADC = 54◦. So ∠EDA = 27◦, and thus φ = 72◦ + 27◦ = 99◦, which is inadmissible, as φ < 90◦. So C is the center of the circle. Then, ∠CAD = ∠CDA = 54◦, ∠ADE = 27◦, and φ = 54◦+27◦ = 81◦.