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 = 6, Perimeter of TUV = 50, Perimeter of XYZ = 150
  • Need to find: length of XY

2. INFER the key relationship

• Since the triangles are similar, their corresponding sides are proportional
• Most importantly: the ratio of their perimeters equals the ratio of any pair of corresponding sides
• This means: \(\frac{\mathrm{Perimeter\ of\ TUV}}{\mathrm{Perimeter\ of\ XYZ}} = \frac{\mathrm{TU}}{\mathrm{XY}}\)

3. TRANSLATE this relationship into an equation

• Set up the proportion: \(\frac{50}{150} = \frac{6}{\mathrm{XY}}\)

4. SIMPLIFY the equation

• First, simplify the left side: \(\frac{50}{150} = \frac{1}{3}\)
• So we have: \(\frac{1}{3} = \frac{6}{\mathrm{XY}}\)
• Cross multiply: \(\mathrm{XY} \times 1 = 6 \times 3\)
• Therefore: \(\mathrm{XY} = 18\)

Answer: C. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the fundamental relationship between perimeter ratios and side ratios in similar triangles. Instead, they might try to work with individual side lengths or set up incorrect proportions like comparing perimeters directly to side lengths without understanding the proportional relationship.

This confusion about the underlying similarity property leads to incorrect setups and may cause them to select Choice B (6) by incorrectly assuming XY equals TU.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might set up the proportion backwards, writing \(\frac{150}{50} = \frac{6}{\mathrm{XY}}\) instead of \(\frac{50}{150} = \frac{6}{\mathrm{XY}}\). This reversal comes from not carefully identifying which triangle is smaller and which is larger.

This leads to the equation \(3 = \frac{6}{\mathrm{XY}}\), giving \(\mathrm{XY} = 2\), causing them to select Choice A (2).

The Bottom Line:

This problem tests whether students truly understand that similarity creates consistent proportional relationships throughout the entire figure - not just between individual corresponding sides, but also between perimeters and other measurements.