In the xy-plane, line ℓ passes through the points \((1, 8)\) and \((5, 2)\). Line m is perpendicular to line...

1. TRANSLATE the problem information

• Given information:
  • Line ℓ passes through points \((1, 8)\) and \((5, 2)\)
  • Line m is perpendicular to line ℓ
  • Need to find the slope of line m

2. INFER the solution strategy

• To find the slope of line m, I first need the slope of line ℓ
• Once I have the slope of line ℓ, I can use the perpendicular lines relationship

3. SIMPLIFY to find the slope of line ℓ

• Apply the slope formula: \(\mathrm{m} = \frac{\mathrm{y_2} - \mathrm{y_1}}{\mathrm{x_2} - \mathrm{x_1}}\)
• \(\mathrm{m_ℓ} = \frac{2 - 8}{5 - 1}\)
  \(= \frac{-6}{4}\)
  \(= -\frac{3}{2}\)

4. INFER the perpendicular relationship

• Perpendicular lines have slopes that are negative reciprocals
• If line ℓ has slope \(-\frac{3}{2}\), then line m has slope: \(-1 \div \left(-\frac{3}{2}\right)\)

5. SIMPLIFY to find the final answer

• \(\mathrm{m_m} = -1 \div \left(-\frac{3}{2}\right)\)
  \(= -1 \times \left(-\frac{2}{3}\right)\)
  \(= \frac{2}{3}\)

Answer: \(\frac{2}{3}\) (Choice D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about perpendicular vs parallel lines: Students remember that related lines have a special slope relationship, but mix up perpendicular and parallel. They think perpendicular lines have the same slope (which is actually true for parallel lines).

This leads them to calculate the slope of line ℓ correctly as \(-\frac{3}{2}\), but then conclude that line m also has slope \(-\frac{3}{2}\), leading them to select Choice A \(\left(-\frac{3}{2}\right)\).

Second Most Common Error:

Weak SIMPLIFY execution with negative reciprocals: Students understand the perpendicular relationship but make errors when computing the negative reciprocal. They might take just the reciprocal (getting \(\frac{2}{3} \rightarrow \frac{3}{2}\)) or just the negative (getting \(-\frac{3}{2} \rightarrow \frac{3}{2}\)) but not both operations correctly.

This may lead them to select Choice E \(\left(\frac{3}{2}\right)\).

The Bottom Line:

This problem tests both computational accuracy and conceptual understanding of perpendicular lines. The key insight is recognizing that 'perpendicular' means negative reciprocal slopes, not the same slopes.