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Perpendicular Lines — Definition, Slopes & Examples

Perpendicular Lines

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Example: Perpendicular Lines

Two straight lines crossing at a right angle (90°), marked with a small square at the intersection point.

 

See also

 

Key Formula

m1×m2=−1m_1 \times m_2 = -1
Where:
  • m1m_1 = Slope of the first line
  • m2m_2 = Slope of the second line

Worked Example

Problem: Determine whether the lines y = 2x + 3 and y = -½x + 1 are perpendicular.
Step 1: Identify the slope of each line from the slope-intercept form y = mx + b.
m1=2,m2=−12m_1 = 2, \quad m_2 = -\tfrac{1}{2}
Step 2: Multiply the two slopes together and check whether the product equals −1.
m1×m2=2×(−12)=−1m_1 \times m_2 = 2 \times \left(-\tfrac{1}{2}\right) = -1
Step 3: Since the product of the slopes is −1, the lines are perpendicular.
Answer: Yes, the two lines are perpendicular because 2 × (−½) = −1.

Why It Matters

Perpendicular lines appear throughout geometry and real life—from the corners of buildings to coordinate axes on a graph. Recognizing perpendicularity lets you construct right angles, prove that shapes are rectangles or squares, and find the shortest distance from a point to a line.

Common Mistakes

Mistake: Confusing perpendicular slopes with equal slopes. Students sometimes think perpendicular lines have the same slope.
Correction: Lines with equal slopes are parallel, not perpendicular. Perpendicular lines have slopes that are negative reciprocals (flip the fraction and change the sign), so their product is −1.

Related Terms

  • Parallel Lines — Lines that never intersect (same slope)
  • Right Angle — The 90° angle formed by perpendicular lines
  • Slope — Measures steepness; key to the perpendicularity test
  • Intersecting Lines — Perpendicular lines are a special case