In the xy-plane, line p has slope -{2/5}. If line q is perpendicular to line p, what is the slope...

1. TRANSLATE the problem information

• Given information:
  • Line p has slope \(-\frac{2}{5}\)
  • Line q is perpendicular to line p
• Find: The slope of line q

2. INFER the key relationship

• Since lines p and q are perpendicular, their slopes must be negative reciprocals
• This means: \((\mathrm{slope\ of\ p}) \times (\mathrm{slope\ of\ q}) = -1\)
• Or equivalently: \(\mathrm{slope\ of\ q} = -\frac{1}{\mathrm{slope\ of\ p}}\)

3. SIMPLIFY to find the slope of line q

• \(\mathrm{slope\ of\ q} = -\frac{1}{\mathrm{slope\ of\ p}}\)
• \(\mathrm{slope\ of\ q} = -\frac{1}{(-\frac{2}{5})}\)
• \(\mathrm{slope\ of\ q} = -1 \times (-\frac{5}{2})\) [dividing by a fraction = multiplying by its reciprocal]
• \(\mathrm{slope\ of\ q} = \frac{5}{2}\)

Answer: D. \(\frac{5}{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not remembering that perpendicular lines have negative reciprocal slopes, thinking instead that perpendicular means 'same slope'

Students might reason: 'If line p has slope \(-\frac{2}{5}\), then line q also has slope \(-\frac{2}{5}\)'

This leads them to select Choice B (\(-\frac{2}{5}\))

Second Most Common Error:

Weak INFER skill: Remembering that perpendicular slopes are reciprocals but forgetting the 'negative' part

Students get the reciprocal of \(-\frac{2}{5}\), which is \(-\frac{5}{2}\), then take the reciprocal to get \(\frac{2}{5}\), missing that it should be the negative reciprocal.

This leads them to select Choice C (\(\frac{2}{5}\))

Third Most Common Error:

Poor SIMPLIFY execution: Understanding the concept correctly but making arithmetic errors with the signs

Students might think 'negative reciprocal' means 'find the reciprocal, then make it negative,' getting reciprocal of \(-\frac{2}{5}\) as \(\frac{5}{2}\), then making it negative to get \(-\frac{5}{2}\).

This leads them to select Choice A (\(-\frac{5}{2}\))

The Bottom Line:

This problem tests both conceptual knowledge of perpendicular line relationships and careful execution of fraction arithmetic with signs. The key insight is that 'negative reciprocal' is a single concept, not two separate operations.