Pentagon PQRST is similar to pentagon UVWXY, such that P, Q, R, S, and T correspond to U, V, W,...

1. TRANSLATE the correspondence information

• Given information:
  • Pentagon PQRST \(\sim\) Pentagon UVWXY
  • P\(\leftrightarrow\)U, Q\(\leftrightarrow\)V, R\(\leftrightarrow\)W, S\(\leftrightarrow\)X, T\(\leftrightarrow\)Y
  • Each side of UVWXY is 3 times its corresponding side in PQRST
  • \(\mathrm{PQ = 8}\)

2. INFER which sides correspond

• Since P corresponds to U and Q corresponds to V
• Side PQ (connecting P to Q) corresponds to side UV (connecting U to V)

3. TRANSLATE the scale factor relationship

• "Each side of UVWXY is 3 times its corresponding side in PQRST"
• This means: \(\mathrm{UV = 3 \times PQ}\)

4. SIMPLIFY to find the answer

• \(\mathrm{UV = 3 \times PQ}\)
  \(\mathrm{UV = 3 \times 8}\)
  \(\mathrm{UV = 24}\)

Answer: D (24)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to connect the vertex correspondence (P\(\leftrightarrow\)U, Q\(\leftrightarrow\)V) with side correspondence (PQ\(\leftrightarrow\)UV). They might think PQ corresponds to a different side like VW or WX, leading to confusion about which relationship to apply.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "3 times the length" and apply the factor incorrectly, perhaps calculating \(\mathrm{PQ \div 3 = 8 \div 3}\) instead of \(\mathrm{PQ \times 3 = 8 \times 3}\). They think the larger pentagon has sides that are 1/3 the size rather than 3 times the size.

This may lead them to select a non-existent fractional answer or cause confusion.

The Bottom Line:

This problem tests whether students understand that correspondence between vertices directly determines correspondence between sides, and whether they can correctly apply scale factors in similar figures. The key insight is that similarity creates a consistent multiplicative relationship between ALL corresponding parts.