Parallelogram ABCD is similar to parallelogram PQRS. The length of each side of parallelogram PQRS is 2 times the length...

1. TRANSLATE the problem information

• Given information:
  • Parallelogram ABCD is similar to parallelogram PQRS
  • Each side of PQRS is 2 times the corresponding side of ABCD
  • Area of ABCD = 5 square centimeters
  • Need to find area of PQRS
• What this tells us: The scale factor between corresponding sides is 2

2. INFER the scaling relationship

• Since we have similar figures and need area, we must apply the area scaling rule
• Key insight: Area doesn't scale the same way as length - it scales by the square of the linear scale factor
• Strategy: Use \(\mathrm{k^2}\) where \(\mathrm{k = 2}\)

3. Apply the area scaling formula

• Linear scale factor: \(\mathrm{k = 2}\)
• Area scale factor: \(\mathrm{k^2 = 2^2 = 4}\)
• Area of PQRS = Area of ABCD × 4 = 5 × 4 = 20
  \(\mathrm{Area\ of\ PQRS = Area\ of\ ABCD \times 4}\)
  \(\mathrm{Area\ of\ PQRS = 5 \times 4}\)
  \(\mathrm{Area\ of\ PQRS = 20}\)

Answer: C. 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly assume area scales linearly with the side lengths

They reason: "If the sides are 2 times longer, then the area is also 2 times larger."

This leads them to calculate: Area of PQRS = \(\mathrm{5 \times 2 = 10}\)

This may lead them to select Choice B (10)

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or apply the \(\mathrm{k^2}\) scaling rule for areas

Without knowing how area scales for similar figures, they might add the scale factor instead of multiplying: \(\mathrm{5 + 2 = 7}\), or make other random calculations that lead to confusion and guessing.

The Bottom Line:

The key challenge is remembering that area scaling follows a square relationship, not a linear one. Students who treat area like a linear dimension will consistently get problems like this wrong.