Line m passes through the points \((0, 3)\) and \((4, 0)\). Line n is perpendicular to line m in the...

1. TRANSLATE the problem information

• Given information:
  • Line m passes through points (0, 3) and (4, 0)
  • Line n is perpendicular to line m
  • Need to find the slope of line n

2. INFER the solution strategy

• To find the slope of line n, we first need the slope of line m
• We can find line m's slope using the two given points
• Then we'll use the perpendicular lines relationship

3. SIMPLIFY to find the slope of line m

• Using the slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Slope of m = \(\mathrm{\frac{0 - 3}{4 - 0} = -\frac{3}{4}}\)

4. INFER and apply the perpendicular relationship

• Perpendicular lines have slopes that are negative reciprocals
• If line m has slope \(\mathrm{-\frac{3}{4}}\), then line n has slope = \(\mathrm{-\frac{1}{-\frac{3}{4}}}\)

5. SIMPLIFY the negative reciprocal calculation

• Slope of n = \(\mathrm{-\frac{1}{-\frac{3}{4}}}\)
• \(\mathrm{= -1 \times (-\frac{4}{3})}\)
• \(\mathrm{= \frac{4}{3}}\)

Answer: D) \(\mathrm{\frac{4}{3}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often confuse which relationship applies to perpendicular lines, incorrectly thinking perpendicular slopes are just negatives of each other (rather than negative reciprocals).

Using this faulty reasoning, they calculate the slope of line m as \(\mathrm{-\frac{3}{4}}\), then think the perpendicular slope is just \(\mathrm{\frac{3}{4}}\). This leads them to select Choice C (\(\mathrm{\frac{3}{4}}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the negative reciprocal of \(\mathrm{-\frac{3}{4}}\), but make an error when handling the double negative in the calculation.

They might calculate: \(\mathrm{-\frac{1}{-\frac{3}{4}} = -\frac{4}{3}}\), forgetting that negative divided by negative gives positive. This causes them to select Choice A (\(\mathrm{-\frac{4}{3}}\)).

The Bottom Line:

This problem tests whether students truly understand the perpendicular lines relationship (negative reciprocals, not just negatives) and can execute the calculation correctly when double negatives are involved.