A square has a side length of s centimeters. The sum of the perimeter of the square and the length...

1. TRANSLATE the problem information

• Given information:
  • Square has side length \(\mathrm{s}\) centimeters
  • Sum of perimeter + one diagonal = \(32 + 8\sqrt{2}\) centimeters
  • Need to find the area

2. INFER what formulas we need

• For a square with side length \(\mathrm{s}\):
  • Perimeter = \(4\mathrm{s}\)
  • Diagonal = \(\mathrm{s}\sqrt{2}\) (creates 45-45-90 right triangles)
  • Area = \(\mathrm{s}^2\)

3. TRANSLATE the constraint into an equation

• "Sum of perimeter and diagonal equals \(32 + 8\sqrt{2}\)"
• Mathematical equation: \(4\mathrm{s} + \mathrm{s}\sqrt{2} = 32 + 8\sqrt{2}\)

4. SIMPLIFY by factoring both sides

• Left side: Factor out \(\mathrm{s}\) → \(\mathrm{s}(4 + \sqrt{2})\)
• Right side: Factor out 8 → \(8(4 + \sqrt{2})\)
• Equation becomes: \(\mathrm{s}(4 + \sqrt{2}) = 8(4 + \sqrt{2})\)

5. SIMPLIFY to solve for \(\mathrm{s}\)

• Divide both sides by \((4 + \sqrt{2})\): \(\mathrm{s} = 8\)

6. INFER the final step

• Question asks for area, not side length
• Area = \(\mathrm{s}^2 = 8^2 = 64\)

Answer: C. 64

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students solve correctly for the side length \(\mathrm{s} = 8\), but then select this as their final answer without recognizing that the question asks for area.

They complete all the algebra correctly but miss the crucial final step of squaring the side length. This leads them to select Choice A (8) instead of calculating \(8^2 = 64\).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that the diagonal of a square equals \(\mathrm{s}\sqrt{2}\), so they set up the equation as \(4\mathrm{s} + \mathrm{s} = 32 + 8\sqrt{2}\) instead.

This gives them \(5\mathrm{s} = 32 + 8\sqrt{2}\), which doesn't factor neatly and leads to confusion with the \(\sqrt{2}\) term. This causes them to get stuck and guess.

The Bottom Line:

This problem combines geometry formulas with algebraic manipulation, then requires careful attention to what the question actually asks for. The clean factoring makes the algebra straightforward, but students must remember both the diagonal formula and to square their result for the area.