Perpendicular Lines
In short: Perpendicular lines meet to form four right angles, and on a coordinate grid the slopes of two such non-vertical lines multiply to negative one.
GeometryGeometryG-GPE.B.5
What is Perpendicular Lines?
Perpendicular lines intersect at a corner of exactly \(90^{\circ}\), and because opposite angles are equal, all four corners at the crossing are right angles. The symbol \(\ell \perp m\) says line \(\ell\) is perpendicular to line \(m\).
In the coordinate plane, a line with slope \(m\) is perpendicular to any line with slope \(-\frac{1}{m}\), the negative reciprocal. A horizontal line and a vertical line are perpendicular too, even though the product rule does not apply to a slope that is undefined.
How to spot it
- A small square is drawn at the point where two lines cross.
- The text uses the word perpendicular, the symbol \(\perp\), or says two lines meet at a right angle; those are perpendicular lines.
- Two slopes are given as fractions that look flipped and have opposite signs, such as \(\frac{3}{5}\) and \(-\frac{5}{3}\).
- A construction asks for the shortest path from a point to a line, such as a hose to a wall.
The method
- Write the slope of the given line, first rearranging the equation into \(y = mx + b\) if needed.
- Flip the fraction and change its sign to get the perpendicular slope.
- Use point-slope form with the required point and the new slope, then simplify.
- Verify by multiplying the two slopes; the product must be \(-1\).
Perpendicular Lines: solved example
Problem
Find the line through \((-2, 4)\) that is perpendicular to \(5x – 3y = 12\).
Solution
Rewrite the given line: \(-3y = -5x + 12\), so \(y = \frac{5}{3}x – 4\). Its slope is \(\frac{5}{3}\), and the negative reciprocal is \(-\frac{3}{5}\).
Point-slope form: \(y – 4 = -\frac{3}{5}(x + 2)\), so \(y = -\frac{3}{5}x – \frac{6}{5} + 4 = -\frac{3}{5}x + \frac{14}{5}\). Product check: \(\frac{5}{3} \times \left(-\frac{3}{5}\right) = -1\). The line is \(y = -\frac{3}{5}x + \frac{14}{5}\).
Try it
Question 1
Are lines with slopes \(\frac{4}{7}\) and \(-\frac{7}{4}\) perpendicular?
Show the answer
Yes: \(\frac{4}{7} \times \left(-\frac{7}{4}\right) = -1\).
Question 2
Are the lines \(y = 2\) and \(x = -5\) perpendicular?
Show the answer
Yes. One is horizontal and the other is vertical, so they meet at a right angle at \((-5, 2)\).
Perpendicular Lines vs. Parallel Lines
Perpendicular pairs cross once, at \(90^{\circ}\), and their slopes are negative reciprocals, like \(\frac{2}{3}\) and \(-\frac{3}{2}\). Parallel pairs never cross and their slopes are identical, like \(\frac{2}{3}\) and \(\frac{2}{3}\). A sign change plus a flip signals a right angle; no change signals a parallel pair.
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