A regular pentagon and a square have the same side length s. The perimeter of the pentagon is 29 centimeters...

1. TRANSLATE the problem information

• Given information:
  • Regular pentagon and square have the same side length \(\mathrm{s}\)
  • Pentagon's perimeter is 29 cm greater than square's perimeter
  • Need to find the value of \(\mathrm{s}\)

2. INFER the approach

• Since both shapes have the same side length \(\mathrm{s}\), I can express both perimeters in terms of \(\mathrm{s}\)
• Pentagon has 5 equal sides → perimeter = \(\mathrm{5s}\)
• Square has 4 equal sides → perimeter = \(\mathrm{4s}\)
• This creates an equation I can solve

3. TRANSLATE the key relationship

• "Pentagon's perimeter is 29 cm greater than square's perimeter" becomes:
  \(\mathrm{5s = 4s + 29}\)

4. SIMPLIFY to find s

• Subtract \(\mathrm{4s}\) from both sides:
  \(\mathrm{5s - 4s = 29}\)
• This gives us: \(\mathrm{s = 29}\)

Answer: C (29)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret the phrase "29 centimeters greater than" and set up the equation incorrectly as \(\mathrm{4s = 5s + 29}\) instead of \(\mathrm{5s = 4s + 29}\).

This reversal leads them to solve:
\(\mathrm{4s - 5s = 29}\)
\(\mathrm{-s = 29}\)
\(\mathrm{s = -29}\)

Since negative side length doesn't make sense, they get confused and may guess randomly or incorrectly assume they made an arithmetic error and select a positive value.

Second Most Common Error:

Poor INFER reasoning: Some students may correctly identify that pentagon perimeter = \(\mathrm{5s}\) and square perimeter = \(\mathrm{4s}\), but then incorrectly think they need to find the actual perimeters first rather than recognizing they can solve directly for \(\mathrm{s}\).

They might try to work backwards from the answer choices, testing each value, which is inefficient and prone to calculation errors. This approach may lead them to select Choice B (25) if they make arithmetic mistakes during their testing process.

The Bottom Line:

This problem requires careful translation of the comparative language ("greater than") into correct mathematical relationships. The key insight is that having a common variable (\(\mathrm{s}\)) allows us to set up and solve an equation directly, rather than needing to find the individual perimeter values first.