*(Question remains unchanged as it was already excellent)* In the coordinate plane, points P and Q have coordinates \((-2, 1)\)...

1. TRANSLATE the problem information

• Given information:
  • Point P has coordinates \((-2, 1)\)
  • Point Q has coordinates \((4, 7)\)
  • Line m is the perpendicular bisector of segment PQ

• What this tells us: Line m is perpendicular to segment PQ (the key insight!)

2. INFER the solution approach

• To find the slope of a perpendicular line, I first need the slope of the original segment PQ
• Then I'll use the perpendicular lines relationship: slopes are negative reciprocals

3. SIMPLIFY to find the slope of segment PQ

• Apply the slope formula: \(\mathrm{slope} = \frac{y_2 - y_1}{x_2 - x_1}\)
• Slope of PQ = \(\frac{7 - 1}{4 - (-2)} = \frac{6}{6} = 1\)

4. INFER the slope of the perpendicular line

• Since perpendicular lines have slopes that are negative reciprocals:
• If PQ has slope 1, then line m has slope \(-\frac{1}{1} = -1\)

Answer: A \((-1)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't fully process what "perpendicular bisector" means and may focus only on finding the midpoint, missing that the line must be perpendicular to PQ.

This leads to confusion about what slope to calculate, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors in the slope calculation, such as computing \(\frac{7-1}{4-2} = \frac{6}{2} = 3\) instead of \(\frac{7-1}{4-(-2)} = \frac{6}{6} = 1\).

When they find the negative reciprocal of 3, they get \(-\frac{1}{3}\), leading them to select Choice C \(-\frac{1}{3}\).

Third Most Common Error:

Conceptual confusion about perpendicular lines: Students correctly find that PQ has slope 1, but either forget that perpendicular lines have negative reciprocal slopes, or remember "reciprocal" but forget the "negative" part.

This may lead them to select Choice E \((1)\).

The Bottom Line:

This problem requires students to connect the geometric concept of "perpendicular bisector" with the algebraic relationship between perpendicular line slopes. Students who can't make this connection, or who make calculation errors, will struggle to reach the correct answer.