In the xy-plane, the line represented by the equation \(\mathrm{y = (-1/4)x + 5}\) is perpendicular to line k. Line...

1. TRANSLATE the problem information

• Given information:
  • Line equation: \(\mathrm{y = (-\frac{1}{4})x + 5}\)
  • Line k is perpendicular to this line
  • Line k passes through points (0, 0) and (c, 12)
• Need to find: value of c

2. INFER what "perpendicular" means mathematically

• Perpendicular lines have slopes that are negative reciprocals
• Strategy: Find the slope of line k first, then use the two points it passes through

3. TRANSLATE the slope from the given line

• From \(\mathrm{y = (-\frac{1}{4})x + 5}\), the slope is \(\mathrm{-\frac{1}{4}}\)
• This means the slope of perpendicular line k is the negative reciprocal

4. SIMPLIFY to find line k's slope

• Negative reciprocal of \(\mathrm{-\frac{1}{4}}\):
• Take reciprocal: \(\mathrm{-\frac{1}{4}}\) becomes \(\mathrm{-\frac{4}{1} = -4}\)
• Take negative: \(\mathrm{-(-4) = 4}\)
• So line k has \(\mathrm{slope = 4}\)

5. INFER the equation of line k

• Line k has slope 4 and passes through origin (0, 0)
• Lines through the origin have equation \(\mathrm{y = mx}\)
• Therefore: \(\mathrm{y = 4x}\)

6. SIMPLIFY to solve for c

• Line k passes through point (c, 12)
• Substitute into \(\mathrm{y = 4x}\): \(\mathrm{12 = 4c}\)
• Solve: \(\mathrm{c = 12 \div 4 = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students know that perpendicular lines have "opposite" slopes but incorrectly think this means just changing the sign, not taking the negative reciprocal.

They might think: "If the slope is \(\mathrm{-\frac{1}{4}}\), then the perpendicular slope is \(\mathrm{+\frac{1}{4}}\)"

This leads them to write \(\mathrm{y = (\frac{1}{4})x}\) for line k, then substitute (c, 12):
\(\mathrm{12 = (\frac{1}{4})c}\), so \(\mathrm{c = 48}\)

This causes confusion when they don't see 48 among typical answer choices and may lead to guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that they need the negative reciprocal of \(\mathrm{-\frac{1}{4}}\), but make arithmetic errors in the calculation.

Common mistake: "\(\mathrm{-\frac{1}{4}}\) becomes \(\mathrm{\frac{1}{4}}\), then negative reciprocal is \(\mathrm{-4}\)"

This gives \(\mathrm{slope = -4}\) for line k, leading to equation \(\mathrm{y = -4x}\)
Substituting (c, 12): \(\mathrm{12 = -4c}\), so \(\mathrm{c = -3}\)

This leads them to select a negative answer if available, or creates confusion about signs.

The Bottom Line:

This problem tests whether students truly understand the relationship between perpendicular slopes (negative reciprocals, not just sign changes) and can execute the reciprocal calculation accurately. The geometric setup is straightforward, but the algebraic precision is crucial.