Triangle ABC is similar to triangle XYZ, where A, B, and C correspond to X, Y, and Z, respectively. In...

1. TRANSLATE the problem information

• Given information:
  • \(\triangle \mathrm{ABC} \sim \triangle \mathrm{XYZ}\) (similar triangles)
  • A corresponds to X, B corresponds to Y, C corresponds to Z
  • \(\mathrm{AB} = 170, \mathrm{BC} = 850, \mathrm{YZ} = 60\)
  • Find: XY

2. INFER which sides correspond

• Since B corresponds to Y and C corresponds to Z, then BC corresponds to YZ
• Since A corresponds to X and B corresponds to Y, then AB corresponds to XY
• Key insight: We have three sides from our triangles (AB, BC, YZ) and need to find the fourth (XY)

3. INFER the proportional relationship

• Similar triangles have proportional corresponding sides
• This means: \(\frac{\mathrm{AB}}{\mathrm{XY}} = \frac{\mathrm{BC}}{\mathrm{YZ}} = \frac{\mathrm{AC}}{\mathrm{XZ}}\)
• We can use the first two ratios since we know AB, BC, and YZ

4. SIMPLIFY by setting up and solving the proportion

• Write the proportion: \(\frac{\mathrm{AB}}{\mathrm{XY}} = \frac{\mathrm{BC}}{\mathrm{YZ}}\)
• Substitute values: \(\frac{170}{\mathrm{XY}} = \frac{850}{60}\)
• Cross multiply: \(170 \times 60 = 850 \times \mathrm{XY}\)
• Calculate: \(10,200 = 850 \times \mathrm{XY}\)
• Solve for XY: \(\mathrm{XY} = 10,200 \div 850 = 12\)

Answer: D. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread the correspondence information and set up incorrect proportions. They might think AB corresponds to YZ instead of XY, leading to the proportion 170/60 = BC/XZ or other incorrect setups. This leads to confusion and incorrect calculations, potentially causing them to select Choice A (204) or abandon systematic solution and guess.

Second Most Common Error:

Poor INFER reasoning: Students recognize the need for proportions but don't properly identify which sides correspond to which. They might randomly pair sides without following the vertex correspondence pattern (A↔X, B↔Y, C↔Z), leading to proportions like XY/BC = YZ/AB. This typically results in getting stuck and selecting Choice C (60) (copying the given YZ value) or Choice B (182).

The Bottom Line:

This problem tests whether students can carefully track correspondence relationships in similar triangles and translate that information into correct proportional setups. Success requires methodical attention to which vertex corresponds to which, not just recognizing that "similar triangles have proportional sides."