A square has a diagonal of length 10 + 5sqrt(2) inches. What is the length, in inches, of one side...

1. TRANSLATE the problem information

• Given information:
  • Square has diagonal length = \(10 + 5\sqrt{2}\) inches
  • Need to find: side length

2. INFER the relationship between diagonal and side

• In a square, we can use the Pythagorean theorem on the right triangle formed by two sides and the diagonal
• If side length = \(\mathrm{s}\), then: \(\mathrm{diagonal}^2 = \mathrm{s}^2 + \mathrm{s}^2 = 2\mathrm{s}^2\)
• Therefore: \(\mathrm{diagonal} = \mathrm{s}\sqrt{2}\)

3. TRANSLATE this relationship into an equation

• We know: \(\mathrm{s}\sqrt{2} = 10 + 5\sqrt{2}\)
• Need to solve for \(\mathrm{s}\): \(\mathrm{s} = \frac{10 + 5\sqrt{2}}{\sqrt{2}}\)

4. SIMPLIFY by rationalizing the denominator

• Multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\): \(\mathrm{s} = \frac{10 + 5\sqrt{2}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\)
• Expand numerator: \(\mathrm{s} = \frac{(10 + 5\sqrt{2})\sqrt{2}}{2}\)
• Distribute: \(\mathrm{s} = \frac{10\sqrt{2} + 5\sqrt{2} \times \sqrt{2}}{2}\)
• Since \(\sqrt{2} \times \sqrt{2} = 2\): \(\mathrm{s} = \frac{10\sqrt{2} + 5 \times 2}{2} = \frac{10\sqrt{2} + 10}{2}\)
• Factor and simplify: \(\mathrm{s} = \frac{10(\sqrt{2} + 1)}{2} = 5(\sqrt{2} + 1) = 5\sqrt{2} + 5 = 5 + 5\sqrt{2}\)

Answer: B \((5 + 5\sqrt{2})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect the diagonal of a square to the Pythagorean theorem. They might know squares have diagonals but fail to derive that \(\mathrm{diagonal} = \mathrm{side} \times \sqrt{2}\). Without this key relationship, they can't set up the fundamental equation \(\mathrm{s}\sqrt{2} = 10 + 5\sqrt{2}\). This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{s} = \frac{10 + 5\sqrt{2}}{\sqrt{2}}\) but make algebraic errors when rationalizing. A common mistake is incorrectly distributing \(\sqrt{2}\) in the numerator, getting \(\frac{10\sqrt{2} + 5\sqrt{2}}{2}\) instead of \(\frac{10\sqrt{2} + 10}{2}\). This calculation error leads them away from the correct answer and toward random selection.

The Bottom Line:

This problem requires connecting geometric relationships (diagonal of square) with algebraic manipulation (rationalizing denominators). Success depends on both deriving the diagonal formula from basic principles and executing multi-step algebraic simplification without computational errors.