Triangle ABC is similar to triangle DEF. The perimeter of triangle DEF is 3/2 times the perimeter of triangle ABC....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Triangle\,ABC} \sim \mathrm{Triangle\,DEF}\) (similar triangles)
  • \(\mathrm{Perimeter\,of\,DEF} = \frac{3}{2} \times \mathrm{Perimeter\,of\,ABC}\)
  • \(\mathrm{Area\,of\,ABC} = 8\,\mathrm{cm}^2\)
  • Need to find: Area of DEF

2. INFER the relationship between perimeter and scale factor

• Since the triangles are similar, all corresponding sides have the same ratio k
• If perimeters have ratio 3/2, then the linear scale factor \(\mathrm{k} = \frac{3}{2}\)
• This is because perimeter scales the same way as individual sides

3. INFER how areas scale in similar figures

• For similar figures, areas scale by the square of the linear scale factor
• Area scale factor = \(\mathrm{k}^2 = \left(\frac{3}{2}\right)^2\)

4. SIMPLIFY to find the area scale factor

• \(\mathrm{k}^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\)

5. SIMPLIFY to find the final area

• Area of DEF = (Area scale factor) × (Area of ABC)
• \(\mathrm{Area\,of\,DEF} = \frac{9}{4} \times 8 = 18\,\mathrm{cm}^2\)

Answer: D (18)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often confuse linear and area scaling relationships. They may think that if the perimeter is 3/2 times larger, then the area is also 3/2 times larger, not recognizing that area scales by \(\mathrm{k}^2\).

This leads them to calculate: \(\mathrm{Area\,of\,DEF} = \frac{3}{2} \times 8 = 12\,\mathrm{cm}^2\)
This may lead them to select Choice A (12)

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret the perimeter relationship, thinking it means DEF has perimeter 3/2 while ABC has perimeter 1, rather than understanding it as a ratio relationship.

This conceptual confusion about what "3/2 times" means can lead to various incorrect calculations and cause them to get stuck and guess randomly.

The Bottom Line:

The key insight is recognizing that linear measurements (like perimeter and side length) scale by k, but area measurements scale by \(\mathrm{k}^2\). Students who miss this fundamental property of similar figures will consistently get area scaling problems wrong.