Triangle ABC and triangle DEF are similar triangles, where AB and DE are corresponding sides. If DE = 2AB and...

1. TRANSLATE the problem information

• Given information:
  • Triangles ABC and DEF are similar
  • AB and DE are corresponding sides
  • \(\mathrm{DE = 2AB}\)
  • \(\mathrm{Perimeter\ of\ triangle\ ABC = 20}\)
• What this tells us: We have a scale factor between the triangles

2. INFER the scale factor relationship

• Since \(\mathrm{DE = 2AB}\) and these are corresponding sides, the scale factor from triangle ABC to triangle DEF is 2
• Key insight: In similar triangles, ALL corresponding sides have the same ratio
• This means every side of triangle DEF is exactly 2 times its corresponding side in triangle ABC

3. INFER how perimeters relate

• If \(\mathrm{DE = 2(AB)}\), \(\mathrm{EF = 2(BC)}\), and \(\mathrm{DF = 2(AC)}\), then:
• \(\mathrm{Perimeter\ of\ DEF = DE + EF + DF}\)

\(\mathrm{Perimeter\ of\ DEF = 2(AB) + 2(BC) + 2(AC)}\)

• Factor out the 2:

\(\mathrm{Perimeter\ of\ DEF = 2(AB + BC + AC)}\)

\(\mathrm{Perimeter\ of\ DEF = 2 \times (Perimeter\ of\ ABC)}\)

• Substitute:

\(\mathrm{Perimeter\ of\ DEF = 2 \times 20 = 40}\)

Answer: B. 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread "\(\mathrm{DE = 2AB}\)" as meaning "\(\mathrm{AB = 2DE}\)" instead

They think triangle ABC is the larger triangle and triangle DEF is smaller, so they use scale factor \(\mathrm{\frac{1}{2}}\) instead of 2. This gives them:

\(\mathrm{Perimeter\ of\ DEF = \frac{1}{2} \times 20 = 10}\)

This may lead them to select Choice A (10).

Second Most Common Error:

Missing conceptual knowledge about similar triangles: Students don't realize that if one pair of corresponding sides has ratio \(\mathrm{2:1}\), then ALL pairs have this same ratio

They might think only DE and AB have this \(\mathrm{2:1}\) relationship, getting confused about how to find the perimeter of triangle DEF without knowing all its side lengths. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students truly understand that similarity means ALL corresponding parts scale by the same factor, not just the parts explicitly mentioned in the problem.