Two similar pentagons have areas in the ratio 4:9. In the smaller pentagon, one side has length 16. What is...

1. TRANSLATE the problem information

• Given information:
  • Two similar pentagons with areas in ratio \(\mathrm{4:9}\)
  • Smaller pentagon has one side of length 16
  • Need to find corresponding side in larger pentagon

2. INFER the key relationship

• Since the figures are similar, their areas scale by \(\mathrm{k^2}\) where \(\mathrm{k}\) is the linear scale factor
• If areas are in ratio \(\mathrm{4:9}\), then linear dimensions are in ratio \(\mathrm{\sqrt{4}:\sqrt{9} = 2:3}\)
• This means every linear measurement (including side lengths) follows the \(\mathrm{2:3}\) ratio

3. INFER the proportion setup

• The ratio of corresponding sides equals the linear ratio
• smaller side/larger side = \(\mathrm{\frac{2}{3}}\)
• Therefore: \(\mathrm{\frac{16}{x} = \frac{2}{3}}\)

4. SIMPLIFY to find the answer

• Cross multiply: \(\mathrm{2x = 48}\)
• Divide both sides by 2: \(\mathrm{x = 24}\)

Answer: C (24)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students use the area ratio directly instead of recognizing they need the linear ratio.

They see areas in ratio \(\mathrm{4:9}\) and immediately set up: \(\mathrm{\frac{16}{x} = \frac{4}{9}}\). Cross multiplying gives \(\mathrm{4x = 144}\), so \(\mathrm{x = 36}\). This leads them to select Choice D (36).

The issue is not recognizing that area scaling and linear scaling are different - areas scale by \(\mathrm{k^2}\) while lengths scale by \(\mathrm{k}\).

The Bottom Line:

This problem tests understanding of how similar figures scale. The key insight is recognizing that area ratios and linear ratios are related by a square relationship. Students who miss this connection will use the wrong ratio and consistently get the wrong answer.