Key takeaway: Perpendicular lines are two lines that intersect at a right angle, and on a coordinate plane their slopes are negative reciprocals, so the product of the two slopes is negative 1.
Grade 10 – GeometryG-GPE.B.5Geometry and Measurement
Definition of Perpendicular Lines
Lines that cross and make a square corner are perpendicular lines. A flagpole standing on level ground, or the edges of a picture frame, show the idea: every one of the four angles at the crossing is \(90^{\circ}\).
Equations reveal it on a grid. A line that climbs 3 for each 4 steps right has a partner at a right angle that drops 4 for each 3 steps right. Level and upright lines are always at right angles, even though the slope rule cannot be applied to an upright one.
Formula and notation
The symbol \(\perp\) means “is perpendicular to.” For slopes \(m_1\) and \(m_2\):
\[ m_1 \cdot m_2 = -1 \quad \text{so} \quad m_2 = -\frac{1}{m_1} \]
A small square drawn at the crossing in a diagram marks the right angle.
Key facts
- Flip the fraction and change the sign to get the slope of a perpendicular line.
- A horizontal line and a vertical line are perpendicular.
- Perpendicular lines form four congruent right angles at their point of intersection.
- The shortest path from a point to a line is the perpendicular segment.
Perpendicular Lines example
A new trail must cross the path \(5x – 2y = 12\) at a right angle and start at \((-3, 7)\). Write its equation.
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Isolate \(y\) in the path: \(-2y = -5x + 12\), so \(y = \frac{5}{2}x – 6\). Slope: \(\frac{5}{2}\).
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Negative reciprocal: \(-\frac{2}{5}\).
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Point-slope at \((-3, 7)\): \(y – 7 = -\frac{2}{5}(x + 3)\), so \(y = -\frac{2}{5}x + \frac{29}{5}\).
The trail is \(y = -\frac{2}{5}x + \frac{29}{5}\).
Common mistake
Watch out
Hugo is told a line has slope \(\frac{3}{8}\) and answers that a perpendicular partner has slope \(\frac{8}{3}\), since he only turned the fraction over.
Why it is wrong
The sign must change as well, giving \(-\frac{8}{3}\). Test by multiplying: \(\frac{3}{8} \cdot (-\frac{8}{3}) = -1\), while \(\frac{3}{8} \cdot \frac{8}{3} = 1\) means the lines are not at right angles.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement