Pentagon P is similar to Pentagon Q. The area of Pentagon P is 9 square centimeters, and the area of...

1. TRANSLATE the problem information

• Given information:
  • Pentagon P similar to Pentagon Q
  • Area of P = \(9\) sq cm, Area of Q = \(81\) sq cm
  • One side of P = \(4\) cm
• What we need to find: corresponding side length of Q

2. INFER the key relationship for similar figures

• When figures are similar, there's a linear scale factor k that relates all corresponding dimensions
• If linear dimensions scale by factor k, then areas scale by factor \(\mathrm{k}^2\)
• This means: \(\frac{\mathrm{Area\;of\;Q}}{\mathrm{Area\;of\;P}} = \mathrm{k}^2\)

3. SIMPLIFY to find the scale factor

• Calculate the area ratio: \(\frac{81}{9} = 9\)
• So \(\mathrm{k}^2 = 9\)
• Taking the square root: \(\mathrm{k = 3}\)
• This means Pentagon Q's sides are 3 times longer than Pentagon P's corresponding sides

4. SIMPLIFY to find the corresponding side length

• Side of Q = k × Side of P
• Side of Q = \(3 \times 4 = 12\) cm

Answer: C (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the area scaling relationship for similar figures. They might think that if areas have ratio \(81:9 = 9:1\), then the sides also have ratio \(9:1\). They calculate \(4 \times 9 = 36\) cm.

This leads them to select Choice D (36).

Second Most Common Error:

Conceptual confusion about scaling: Students might incorrectly think the side ratio equals the area ratio directly, or they might take the area ratio and divide instead of taking square root. They calculate \(\frac{81}{9} = 9\), then try \(4 \times \frac{9}{4} = 9\) or similar incorrect operations.

This may lead them to select Choice B (9).

The Bottom Line:

The key insight is recognizing that linear and area scaling follow different rules - you must take the square root of the area ratio to get the linear scale factor. Missing this relationship makes systematic solution impossible.