Triangles ABC and DEF are similar. The perimeter of triangle DEF is 3 times the perimeter of triangle ABC. If...

1. TRANSLATE the problem information

• Given information:
  • Triangles ABC and DEF are similar
  • Perimeter of DEF = \(\mathrm{3 \times perimeter\,of\,ABC}\)
  • Area of ABC = \(\mathrm{18}\) square centimeters
  • Need to find: Area of DEF

2. INFER the scaling relationship

• Since the triangles are similar, the perimeter ratio tells us the linear scale factor
• Linear scale factor from ABC to DEF = \(\mathrm{3}\)
• Key insight: Areas of similar figures scale by the SQUARE of the linear scale factor
• Area scale factor = \(\mathrm{3^2 = 9}\)

3. SIMPLIFY to find the final answer

• Area of DEF = Area of ABC \(\mathrm{\times}\) area scale factor
• Area of DEF = \(\mathrm{18 \times 9 = 162}\) square centimeters

Answer: C (162)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students apply the linear scale factor directly to the area instead of squaring it first.

They think: "If the perimeter is 3 times larger, then the area is also 3 times larger." This reasoning leads to: Area of DEF = \(\mathrm{18 \times 3 = 54}\).

This may lead them to select Choice B (54).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "3 times the perimeter" as meaning they should divide instead of multiply.

They might think the area of ABC is larger than DEF, calculating: Area of DEF = \(\mathrm{18 \div 3 = 6}\).

This may lead them to select Choice A (6).

The Bottom Line:

The key challenge is remembering that while linear measurements scale by factor k, areas scale by \(\mathrm{k^2}\). This is a fundamental property of similar figures that students often forget under test pressure, defaulting to applying the same scale factor to everything.