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Perpendicular

Perpendicular
Normal
Orthogonal

At a 90° angle. Note: Perpendicular lines have slopes that are negative reciprocals.

 

Example: Perpendicular Lines

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

 

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: Line A passes through the points (1, 2) and (3, 6). Line B passes through the points (0, 4) and (4, 2). Determine whether Line A and Line B are perpendicular.
Step 1: Find the slope of Line A using the slope formula.
mA=6−23−1=42=2m_A = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
Step 2: Find the slope of Line B using the slope formula.
mB=2−44−0=−24=−12m_B = \frac{2 - 4}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}
Step 3: Multiply the two slopes together. If the product equals −1, the lines are perpendicular.
mA×mB=2×(−12)=−1m_A \times m_B = 2 \times \left(-\frac{1}{2}\right) = -1
Answer: Since the product of the slopes is −1, Line A and Line B are perpendicular.

Another Example

Problem: Find the slope of a line perpendicular to the line y = 3x + 5.
Step 1: Identify the slope of the given line from its equation in slope-intercept form.
m=3m = 3
Step 2: Take the negative reciprocal to find the perpendicular slope. Flip the fraction and change the sign.
m⊥=−13m_{\perp} = -\frac{1}{3}
Answer: Any line with slope −1/3 is perpendicular to y = 3x + 5.

Frequently Asked Questions

What is the difference between perpendicular and parallel lines?
Perpendicular lines intersect at exactly 90°, while parallel lines never intersect and stay the same distance apart. Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals of each other.
Can a horizontal line be perpendicular to a vertical line?
Yes. A horizontal line (slope = 0) and a vertical line (undefined slope) meet at a right angle, so they are perpendicular. The negative reciprocal rule doesn't apply directly here because the vertical line's slope is undefined, but the 90° angle definition still holds.

Perpendicular lines vs. Parallel lines

Perpendicular lines cross at a 90° angle, with slopes whose product is −1. Parallel lines never cross and have identical slopes. Both concepts describe a specific angle relationship between two lines: perpendicular means 90°, parallel means 0° (no rotation between them).

Why It Matters

Perpendicularity appears throughout geometry, construction, and design. Builders use right angles to ensure walls are straight, and engineers rely on perpendicular forces when analyzing structures. In coordinate geometry, the negative reciprocal slope relationship lets you write equations of altitude lines in triangles, construct perpendicular bisectors, and find shortest distances from points to lines.

Common Mistakes

Mistake: Taking only the reciprocal of the slope without changing the sign (e.g., thinking the perpendicular slope of 2 is 1/2).
Correction: You must both flip the fraction and negate it. The perpendicular slope of 2 is −1/2, not 1/2. Always check that the product of the two slopes equals −1.
Mistake: Assuming two lines that intersect must be perpendicular.
Correction: Two lines can intersect at any angle. They are only perpendicular if that angle is exactly 90°. Verify by checking whether the slopes multiply to −1 or by measuring the angle directly.

Related Terms

  • Angle — Perpendicular lines form a 90° angle
  • Line — The geometric object that can be perpendicular
  • Slope of a Line — Used to test for perpendicularity
  • Negative Reciprocal — Slope relationship between perpendicular lines
  • Parallel — Lines that never intersect, contrasted with perpendicular
  • Right Angle — The 90° angle formed by perpendicular lines
  • Perpendicular Bisector — A line perpendicular to a segment at its midpoint