Rectangle ABCD has length 12 and width 8. Rectangle PQRS has length 18. Which additional piece of information is sufficient...

1. TRANSLATE the problem information

• Given information:
  • Rectangle ABCD: length = 12, width = 8
  • Rectangle PQRS: length = 18, width = unknown
• What we need: sufficient information to prove similarity

2. INFER the similarity requirement

• Key insight: For rectangles to be similar, ratios of corresponding sides must be equal
• Since we know both lengths, we can find the length ratio first
• Length ratio = \(\frac{12}{18} = \frac{2}{3}\)

3. INFER what the width ratio must be

• For similarity: width ratio must also equal \(\frac{2}{3}\)
• If width of PQRS is w, then: \(\frac{8}{w} = \frac{2}{3}\)
• Solving: \(w = 8 \times \frac{3}{2} = 12\)

4. SIMPLIFY by checking each answer choice

Let's systematically verify each option:

(A) Width of PQRS is 12:

• Width ratio = \(\frac{8}{12} = \frac{2}{3}\) ✓
• This matches our length ratio!

(B) Width of PQRS is 15:

• Width ratio = \(\frac{8}{15} \neq \frac{2}{3}\) ✗

(C) Area of PQRS is 200:

• Width = Area ÷ length = \(\frac{200}{18} \approx 11.11\) (use calculator)
• Width ratio = \(\frac{8}{11.11} \neq \frac{2}{3}\) ✗

(D) Diagonal of PQRS is 25:

• Using Pythagorean theorem: \(18^2 + \mathrm{width}^2 = 25^2\)
• \(\mathrm{width}^2 = 625 - 324 = 301\)
• \(\mathrm{width} = \sqrt{301} \approx 17.35\) (use calculator)
• Width ratio = \(\frac{8}{17.35} \neq \frac{2}{3}\) ✗

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that similarity requires proportional corresponding sides, instead thinking that 'similar' just means having the same shape or some vague resemblance.

Without understanding the proportionality requirement, they might pick any answer that 'sounds reasonable' or guess randomly. This leads to confusion and guessing rather than systematic analysis.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which sides correspond to which when setting up ratios, possibly comparing length of one rectangle to width of another.

For example, they might incorrectly set up \(\frac{12}{8} = \frac{18}{w}\), thinking they're comparing 'first dimension to second dimension' rather than 'length to length, width to width.' This leads them to calculate w = 12, making them select Choice A (12) but for the wrong mathematical reasoning.

The Bottom Line:

This problem requires understanding that geometric similarity has a precise mathematical definition involving proportional corresponding sides, not just visual resemblance.