Version 1: Current SPR Format (Recommended)The area of a square is 441 square inches. The length of a diagonal of...

1. TRANSLATE the problem information

• Given information:
  • Area of square = \(\mathrm{441\, square\, inches}\)
  • Diagonal length = \(\mathrm{d\sqrt{2}\, inches}\) (where d is unknown)
• Need to find: the value of d

2. INFER the solution approach

• To find the diagonal, I first need the side length of the square
• I can get the side length from the area using \(\mathrm{A = s^2}\)
• Then I'll use the geometric relationship between side and diagonal

3. SIMPLIFY to find the side length

• From \(\mathrm{A = s^2}\): \(\mathrm{s^2 = 441}\)
• Taking the square root: \(\mathrm{s = \sqrt{441} = 21\, inches}\)

4. INFER the diagonal relationship

• In a square, the diagonal creates two right triangles
• Each triangle has legs of length s and hypotenuse equal to the diagonal
• Using Pythagorean theorem: \(\mathrm{diagonal^2 = s^2 + s^2 = 2s^2}\)
• Therefore: \(\mathrm{diagonal = s\sqrt{2}}\)

5. SIMPLIFY to find d

• Diagonal = \(\mathrm{21\sqrt{2}\, inches}\) (substituting \(\mathrm{s = 21}\))
• But we're told diagonal = \(\mathrm{d\sqrt{2}\, inches}\)
• Setting these equal: \(\mathrm{d\sqrt{2} = 21\sqrt{2}}\)
• Therefore: \(\mathrm{d = 21}\)

Answer: 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about area vs. perimeter: Students might use the perimeter formula instead of area formula, thinking "\(\mathrm{441 = 4s}\)" and getting \(\mathrm{s = 110.25}\), leading to an unreasonably large diagonal value. This leads to confusion and guessing.

Second Most Common Error:

Weak INFER skill: Students may find the side length correctly (\(\mathrm{s = 21}\)) but not know how to find the diagonal of a square. Without recognizing that \(\mathrm{diagonal = side\sqrt{2}}\), they get stuck after finding \(\mathrm{s = 21}\) and resort to guessing.

The Bottom Line:

This problem tests whether students can work backwards from area to find side length, then apply geometric properties to find the diagonal. The key insight is recognizing that you need to find the side first before you can determine the diagonal relationship.