Square X has a side length of 12 centimeters. The perimeter of square Y is half the area of square...

1. TRANSLATE the problem information

• Given information:
  • Square X has side length = 12 cm
  • Perimeter of square Y = half the area of square X
  • Find: side length of square Y

• What this tells us: We need to work with square X first, then use that result for square Y

2. INFER the solution approach

• We need area of square X to find perimeter of square Y
• Then we'll use the perimeter of square Y to find its side length
• This is a multi-step problem requiring two different square formulas

3. SIMPLIFY to find area of square X

• Area of square X = \(\mathrm{side}^2\)
• Area = \(12^2\)
  Area = \(12 \times 12\)
  Area = \(144\) square centimeters

4. TRANSLATE to find perimeter of square Y

• Perimeter of square Y = half the area of square X
• Perimeter of square Y = \(144 \div 2\)
  Perimeter of square Y = \(72\) centimeters

5. INFER how to find side length from perimeter

• For any square: \(\mathrm{Perimeter} = 4 \times \mathrm{side\ length}\)
• Therefore: \(\mathrm{side\ length} = \mathrm{Perimeter} \div 4\)

6. SIMPLIFY to find the final answer

• Side length of square Y = \(72 \div 4\)
  Side length of square Y = \(18\) centimeters

Answer: 18 centimeters (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misunderstanding "half the area" and instead thinking the perimeter of square Y equals the full area of square X

Students read "The perimeter of square Y is half the area of square X" but mentally process it as "The perimeter of square Y is the area of square X," skipping the "half" part. They then calculate: perimeter of Y = \(144\), so side length = \(144 \div 4\)
side length = \(36\).

This may lead them to select Choice D (36)

Second Most Common Error:

Conceptual confusion about the relationship: Thinking that "half the area" means half the side length rather than half the numerical value of the area

Students might reason: "If it's half the area, then the side must be half of 12." They calculate side length of Y = \(12 \div 2\)
side length of Y = \(6\), completely bypassing the area and perimeter calculations.

This may lead them to select Choice A (6)

The Bottom Line:

This problem challenges students to carefully TRANSLATE a complex relationship between different measurements (area → perimeter → side length) while keeping track of which square they're working with at each step.