In the xy-plane, a line ell passes through the point \((-7, -7)\). The shortest distance from the point \((-4, -6)\)...

1. TRANSLATE the problem information

• Given information:
  • Line \(\ell\) passes through point \((-7, -7)\)
  • The shortest distance from point \((-4, -6)\) to line \(\ell\) occurs at \((-7, -7)\)

• What this tells us: The point \((-7, -7)\) is where the perpendicular from \((-4, -6)\) meets line \(\ell\)

2. INFER the geometric relationship

• Key insight: The shortest distance from any point to a line is always along the perpendicular to that line
• This means the segment from \((-4, -6)\) to \((-7, -7)\) is perpendicular to line \(\ell\)
• Strategy: Find the slope of this perpendicular segment, then use the negative reciprocal property

3. SIMPLIFY to find the perpendicular segment's slope

• Using the slope formula with points \((-4, -6)\) and \((-7, -7)\):
  \(\mathrm{m_{segment}} = \frac{\mathrm{y_2 - y_1}}{\mathrm{x_2 - x_1}}\)
  \(= \frac{-7 - (-6)}{-7 - (-4)}\)
  \(= \frac{-1}{-3}\)
  \(= \frac{1}{3}\)

4. INFER the slope of line \(\ell\)

• Since perpendicular lines have slopes that are negative reciprocals:
• If the perpendicular segment has slope \(\frac{1}{3}\), then line \(\ell\) has slope \(-\frac{1}{\frac{1}{3}} = -3\)

Answer: (A) -3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that "shortest distance occurs at \((-7, -7)\)" means the segment from \((-4, -6)\) to \((-7, -7)\) is perpendicular to the line.

Students might think they need to use the distance formula or try to work with the equation of line \(\ell\) directly. Without recognizing the perpendicular relationship, they cannot access the negative reciprocal property that makes this problem solvable. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Getting the arithmetic wrong when calculating the slope or finding the negative reciprocal.

Common mistakes include: calculating \(\frac{-1}{-3}\) as \(-\frac{1}{3}\) instead of \(\frac{1}{3}\), or finding the negative reciprocal of \(\frac{1}{3}\) as \(\frac{1}{3}\) instead of \(-3\). Depending on the specific error, this may lead them to select Choice (B) \((-\frac{1}{3})\) or Choice (C) \((\frac{1}{3})\).

The Bottom Line:

This problem tests whether students can connect the geometric concept of "shortest distance" to the algebraic concept of perpendicular slopes. The key breakthrough is realizing that the problem has already given you the foot of the perpendicular - you just need to use it.