The table gives the perimeters of similar triangles TUV and XYZ, where TU corresponds to XY. The length of TU...

1. TRANSLATE the problem information

• Given information:
  • Triangle TUV is similar to Triangle XYZ
  • TU corresponds to XY
  • TU = 18
  • Perimeter of TUV = 37
  • Perimeter of XYZ = 333
  • Find: length of XY

2. INFER the key relationship

• Since the triangles are similar, all corresponding sides are proportional by the same scale factor k
• This means: \(\mathrm{XY = k \times TU = k \times 18}\)
• Key insight: The perimeters are also proportional by this same scale factor k

3. INFER the strategy to find the scale factor

• We can find k using the perimeter ratio since we know both perimeters
• Perimeter of XYZ = k × Perimeter of TUV
• \(333 = k \times 37\)

4. SIMPLIFY to find the scale factor

• \(\mathrm{k = 333 \div 37 = 9}\)

5. SIMPLIFY to find XY

• \(\mathrm{XY = k \times TU = 9 \times 18 = 162}\)

Answer: D. 162

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they can use the perimeter ratio to find the scale factor. Instead, they might try to work directly with individual sides or get confused about how to start the problem. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students may confuse which triangle is which, or think that TU and XY have the same length because they correspond. This may lead them to select Choice B (18).

The Bottom Line:

The key insight is recognizing that similar triangles have ALL measurements (sides, perimeters, etc.) proportional by the same scale factor. Once you find this factor using any known pair of corresponding measurements, you can apply it to find any unknown corresponding measurement.