Rectangle P and rectangle Q are similar. The area of rectangle Q is 9 times the area of rectangle P....

1. TRANSLATE the problem information

• Given information:
  • Rectangle P and Q are similar
  • \(\mathrm{Area\ of\ Q = 9 \times Area\ of\ P}\)
  • \(\mathrm{Perimeter\ of\ P = 12}\)
  • Need to find: Perimeter of Q

• What this tells us: We have an area relationship and need to find a perimeter relationship.

2. INFER the scaling relationship

• Key insight: For similar figures, there's a consistent scaling factor \(\mathrm{k}\)
• If corresponding sides differ by factor \(\mathrm{k}\), then:
  • Perimeters also differ by factor \(\mathrm{k}\)
  • Areas differ by factor \(\mathrm{k^2}\)
• Since we know the area relationship, we can work backwards to find \(\mathrm{k}\)

3. SIMPLIFY to find the linear scaling factor

• We know: Area ratio = \(\mathrm{k^2 = 9}\)
• To find \(\mathrm{k}\): Take the square root of both sides
• \(\mathrm{k = \sqrt{9} = 3}\)
• This means corresponding sides (and perimeters) have ratio \(\mathrm{3:1}\)

4. INFER and calculate the final answer

• Since \(\mathrm{\frac{Perimeter\ of\ Q}{Perimeter\ of\ P} = k = 3}\)
• \(\mathrm{Perimeter\ of\ Q = 3 \times Perimeter\ of\ P}\)
• \(\mathrm{Perimeter\ of\ Q = 3 \times 12 = 36}\)

Answer: B. 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often confuse area scaling with perimeter scaling, thinking that if the area is 9 times larger, the perimeter is also 9 times larger.

They miss the crucial relationship that area scales by \(\mathrm{k^2}\) while perimeter scales by \(\mathrm{k}\). This leads them to calculate: \(\mathrm{Perimeter\ of\ Q = 9 \times 12 = 108}\).

This may lead them to select Choice D (108).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students understand the \(\mathrm{k^2}\) relationship but make calculation errors when finding the square root or in the final multiplication.

Some students might incorrectly calculate \(\mathrm{\sqrt{9}}\) or make arithmetic errors in the final step, potentially leading to wrong answer choices or confusion and guessing.

The Bottom Line:

The key challenge is recognizing that different geometric properties scale differently in similar figures - areas scale quadratically (\(\mathrm{k^2}\)) while linear measurements like perimeter scale linearly (\(\mathrm{k}\)). Students must bridge from the given area relationship to find the linear scaling factor first.