In the xy-plane, line m is defined by the equation 2x - 5y = 10. Line p is the perpendicular...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the xy-plane, line m is defined by the equation \(2\mathrm{x} - 5\mathrm{y} = 10\). Line p is the perpendicular from the origin \((0,0)\) to line m. What is the slope of line p?
1. TRANSLATE the given equation to find the slope
- Given: Line m has equation \(\mathrm{2x - 5y = 10}\)
- To find the slope, convert to slope-intercept form \(\mathrm{y = mx + b}\):
- \(\mathrm{2x - 5y = 10}\)
- \(\mathrm{-5y = -2x + 10}\)
- \(\mathrm{y = \frac{2}{5}x - 2}\)
- The slope of line m is \(\mathrm{\frac{2}{5}}\)
2. INFER the relationship for perpendicular lines
- Line p is perpendicular to line m
- Key insight: Perpendicular lines have slopes that are negative reciprocals
- If one line has slope m, the perpendicular line has slope \(\mathrm{-\frac{1}{m}}\)
3. SIMPLIFY to find the perpendicular slope
- Slope of line p = \(\mathrm{-\frac{1}{\text{slope of line m}}}\)
- Slope of line p = \(\mathrm{-\frac{1}{\frac{2}{5}}}\)
\(\mathrm{= -1 \times \frac{5}{2}}\)
\(\mathrm{= -\frac{5}{2}}\)
Answer: \(\mathrm{-\frac{5}{2}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may incorrectly rearrange the equation or make sign errors when converting to slope-intercept form. For instance, they might get \(\mathrm{y = -\frac{2}{5}x - 2}\), thinking the slope of line m is \(\mathrm{-\frac{2}{5}}\). Then applying the perpendicular relationship: \(\mathrm{-\frac{1}{-\frac{2}{5}} = \frac{5}{2}}\).
This may lead them to select Choice (D) \(\mathrm{\frac{5}{2}}\).
Second Most Common Error:
Poor INFER reasoning: Students remember that perpendicular lines are related but confuse the relationship. They might think perpendicular slopes are just the opposite sign (not reciprocal), so they'd say the perpendicular slope is \(\mathrm{-\frac{2}{5}}\).
This may lead them to select Choice (B) \(\mathrm{-\frac{2}{5}}\).
The Bottom Line:
This problem tests whether students can systematically work with linear equations and apply the perpendicular slope relationship correctly. Success requires careful algebraic manipulation followed by precise application of the negative reciprocal rule.