Practice: Which Quad?

The Puzzle (Page 111)

Instructions:

1. Fold a sheet into half.
2. Fold it once more into a quarter.
3. Make a triangular crease at the corner that is at the middle of the paper.
4. Open the sheet.

Analysis:

• The “middle of the paper” corresponds to the center of the unfolded sheet.
• Folding into quarters creates 4 layers.
• A single straight crease across the corner of the folded quarter cuts through all 4 layers symmetrically.
• When unfolded, this single line segment is reflected across the horizontal and vertical axes.

Result: The shape formed is a Rhombus.

• The diagonals of the unfolded shape lie along the fold lines.
• Since the paper was folded perpendicular to the sides, the diagonals bisect each other at $90^\circ$.
• Since the cut was straight, the resulting sides connecting the axes are straight lines forming a rhombus.

Variations:

• To get a Square: You must fold the quarter such that the crease makes a $45^\circ$ angle with the fold edge (making an isosceles triangle at the corner). This ensures the diagonals are equal in length.
• To get a Kite: Fold differently so adjacent sides are equal but the shape lacks the full symmetry of a rhombus.