In the xy-plane, line k is the perpendicular bisector of the line segment with endpoints at \((2, 9)\) and \((8,...

1. TRANSLATE the problem information

• Given information:
  • Line k is the perpendicular bisector of a line segment
  • The line segment has endpoints (2, 9) and (8, 1)
  • We need to find the slope of line k
• What this tells us: We need to find the slope of the original line segment, then find the slope of a line perpendicular to it.

2. INFER the approach

• Since line k is perpendicular to the line segment, its slope will be the negative reciprocal of the segment's slope
• Strategy: First find the slope of the segment, then calculate its negative reciprocal

3. SIMPLIFY to find the segment's slope

• Apply the slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute the coordinates: \(\mathrm{m = \frac{1 - 9}{8 - 2}}\)
• Calculate: \(\mathrm{m = \frac{-8}{6} = \frac{-4}{3}}\)

4. SIMPLIFY to find the perpendicular slope

• The slope of line k is the negative reciprocal of \(\mathrm{\frac{-4}{3}}\)
• Calculate: \(\mathrm{m_k = -1 \div (\frac{-4}{3}) = -1 \times (\frac{-3}{4}) = \frac{3}{4}}\)

Answer: 3/4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make calculation errors when finding the negative reciprocal of \(\mathrm{\frac{-4}{3}}\). They might calculate \(\mathrm{-1 \div (\frac{-4}{3})}\) incorrectly, getting \(\mathrm{\frac{-3}{4}}\) instead of \(\mathrm{\frac{3}{4}}\), or make errors in the division process itself.

This leads to selecting an incorrect answer or getting confused and guessing.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that perpendicular lines have slopes that are negative reciprocals. They might think perpendicular slopes are just negatives of each other, leading them to calculate the slope as \(\mathrm{\frac{4}{3}}\) instead of \(\mathrm{\frac{3}{4}}\).

This causes them to arrive at the wrong answer systematically.

The Bottom Line:

This problem tests both conceptual understanding of perpendicular relationships and careful arithmetic execution. Students need to correctly apply the perpendicular slope relationship AND execute the negative reciprocal calculation without errors.