Triangle ABC is similar to triangle DEF. The length of each side of triangle DEF is 3 times the length...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC is similar to triangle DEF
  • Each side of triangle DEF is 3 times its corresponding side in triangle ABC
  • Area of triangle ABC = 4 square cm
  • Need to find: Area of triangle DEF

2. INFER the scaling relationship

• Key insight: When figures are similar, their areas don't scale the same way as their sides
• If linear dimensions scale by factor k, then areas scale by factor \(\mathrm{k^2}\)
• Since each side of DEF is 3 times the corresponding side of ABC, our linear scale factor is \(\mathrm{k = 3}\)

3. SIMPLIFY to find the area scale factor

• Area scale factor = \(\mathrm{k^2}\) = \(\mathrm{3^2}\) = \(\mathrm{9}\)
• This means triangle DEF has 9 times the area of triangle ABC

4. Calculate the final answer

• Area of triangle DEF = Area of triangle ABC × 9
• Area of triangle DEF = \(\mathrm{4 \times 9}\) = \(\mathrm{36}\) square cm

Answer: D (36)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students apply the linear scale factor directly to area, thinking that if sides are 3 times larger, then area is also 3 times larger.

They calculate: Area of DEF = \(\mathrm{4 \times 3}\) = \(\mathrm{12}\) square cm

This leads them to select Choice B (12).

Second Most Common Error:

Conceptual confusion about scaling: Students might try to add the scale factor instead of multiplying, or get confused about which operation to perform.

Some students calculate: Area of DEF = \(\mathrm{4 + 3}\) = \(\mathrm{7}\) square cm, leading to Choice A (7).

Others might calculate \(\mathrm{4 \times 4}\) = \(\mathrm{16}\) by mistakenly using \(\mathrm{k + 1 = 4}\) as their factor, leading to Choice C (16).

The Bottom Line:

The key insight that separates successful students from struggling ones is understanding that area scaling follows a \(\mathrm{k^2}\) rule, not a k rule. This requires connecting the concept of similar figures with the geometric principle of how two-dimensional measurements scale differently than one-dimensional measurements.