2x + y = 37 In triangle QRS, sides QR and RS each have a length of x centimeters and...

1. TRANSLATE the problem information

• Given information:
  • Triangle QRS has two sides (QR and RS) each with length \(\mathrm{x}\) centimeters
  • The third side (SQ) has length \(\mathrm{y}\) centimeters
  • The equation \(\mathrm{2x + y = 37}\) represents this situation

• What this tells us: We need to figure out what the expression \(\mathrm{2x + y}\) represents in terms of the triangle

2. INFER what the expression \(\mathrm{2x + y}\) represents

• Since \(\mathrm{QR = x}\) and \(\mathrm{RS = x}\), the sum of these two equal sides is \(\mathrm{x + x = 2x}\)
• The third side \(\mathrm{SQ = y}\)
• Therefore: \(\mathrm{2x + y}\) = (sum of the two equal sides) + (third side) = sum of all three sides
• This means \(\mathrm{2x + y}\) represents the perimeter of triangle QRS

3. INFER what 37 represents

• Since we established that \(\mathrm{2x + y}\) = perimeter of the triangle
• And we're told that \(\mathrm{2x + y = 37}\)
• Therefore: 37 = perimeter of the triangle
• The perimeter is the sum of the lengths of all three sides

Answer: C. The sum of the lengths, in centimeters, of the three sides of the triangle

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may focus on individual components (like \(\mathrm{x}\) or \(\mathrm{y}\)) rather than understanding what the entire expression \(\mathrm{2x + y}\) represents in the geometric context.

For example, they might see "\(\mathrm{2x}\)" and think it only relates to the two equal sides, or see "\(\mathrm{y}\)" and focus only on the third side, without recognizing that \(\mathrm{2x + y}\) represents the combination of all sides. This leads to confusion about what 37 could represent and may cause them to guess among the answer choices.

The Bottom Line:

This problem tests whether students can connect algebraic expressions to their geometric meaning. The key insight is recognizing that \(\mathrm{2x + y}\) isn't just a mathematical expression—it represents a specific geometric quantity (the perimeter) in the context of the triangle.