Exploring Ratios in Paper-Folding Geometry: A Challenge from the Australian Math Competition

In this problem, we investigate how folding a rectangle with a \(3:1\) length-to-width ratio along its diagonal creates a pentagon, then calculate the ratio of the pentagon’s area to the rectangle’s area. Here’s a step-by-step breakdown:

Problem Overview:

• Given: A rectangle with length \(3x\) and width \(x\).
• Goal: Find the area ratio of the pentagon (formed by folding along the diagonal) to the original rectangle.

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Key Concepts:

1. Diagonal Folding: Folding divides the rectangle into two congruent triangles. Upon folding, a pentagon is formed with overlapping areas.
2. Using Pythagorean Theorem: The length of the diagonal \(d\) is:
   \(d = \sqrt{(3x)^2 + x^2} = x \sqrt{10}\)

Solution:

To calculate the pentagon’s area, we:

1. Divide Areas: Recognize that the folded area includes parts of two triangles and involves subtracting a duplicated section.
2. Area Calculation:

• Rectangle: \(A_{\text{rect}} = 3x^2 \)
• Pentagon: Using geometric transformations and symmetry, we find the final area of the pentagon, which simplifies to a specific proportion of the rectangle.

This problem illustrates the beauty of geometric folding and similarity concepts, sharpening skills in spatial reasoning and area transformations.