Rectangle R has length 10 centimeters and width 6 centimeters. Rectangle S is similar to rectangle R, with perimeter 150...

1. TRANSLATE the problem information

• Given information:
  • Rectangle R: length = 10 cm, width = 6 cm
  • Rectangle S is similar to rectangle R
  • Perimeter of S = 150% of perimeter of R
  • Need to find: length of rectangle S

• What this tells us: \(\mathrm{P_S = 1.5 \times P_R}\)

2. SIMPLIFY to find rectangle R's perimeter

• \(\mathrm{P_R = 2(length + width)}\)
  \(\mathrm{= 2(10 + 6)}\)
  \(\mathrm{= 2(16)}\)
  \(\mathrm{= 32\,cm}\)

3. SIMPLIFY to find rectangle S's perimeter

• \(\mathrm{P_S = 1.5 \times 32}\)
  \(\mathrm{= 48\,cm}\)

4. INFER the scaling relationship for similar figures

• Key insight: Since rectangles R and S are similar, ALL linear dimensions scale by the same factor
• If the perimeter increased by factor 1.5, then each linear dimension also increases by factor 1.5
• This is because perimeter is the sum of linear dimensions

5. SIMPLIFY to find the length of rectangle S

• Length of S = (scale factor) × (length of R)
• \(\mathrm{Length\,of\,S = 1.5 \times 10}\)
  \(\mathrm{= 15\,cm}\)

Answer: C. 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect perimeter scaling to linear dimension scaling for similar figures.

They correctly find that rectangle S has perimeter 48 cm, but then try to work backwards using the constraint that rectangles are similar without recognizing the direct scaling relationship. This leads them to set up more complex systems of equations or get confused about how to use the similarity condition effectively.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about similarity: Students misunderstand what "similar" means in the context of rectangles.

Some students think "similar rectangles" just means rectangles with the same shape, but don't realize this means ALL corresponding linear dimensions are proportional by the same ratio. They might try to find a rectangle with perimeter 48 that has some relationship to the 10×6 rectangle, but without using the proper scaling factor.

This may lead them to select Choice B (10) by incorrectly thinking the length stays the same.

The Bottom Line:

The key insight is recognizing that for similar figures, there's a direct scaling relationship between corresponding measurements. If you know how one linear measurement scales, you know how ALL linear measurements scale.