Perpendicular Lines [H]

Perpendicular Lines

Two lines are perpendicular if they meet at a right angle (90°). The gradients of perpendicular lines are related by a simple rule: if one line has gradient \( m \), any perpendicular line has gradient \( -\dfrac{1}{m} \) (the negative reciprocal). Equivalently:

\[ m_1 \times m_2 = -1 \]

Examples: gradient 3 is perpendicular to gradient \( -\frac{1}{3} \); gradient \( -\frac{2}{5} \) is perpendicular to gradient \( \frac{5}{2} \). A horizontal line (gradient 0) is perpendicular to a vertical line (undefined gradient).

x y 0 −2 2 4 6 4 2 −2 y = 2x − 1 y = −½x + 3 m₁ × m₂ = 2 × (−½) = −1 ✓

Worked Examples

State the gradient of a line perpendicular to each: \( m = 4 \), \( m = -\frac{2}{3} \), \( m = \frac{1}{5} \).

Perpendicular gradient = negative reciprocal.

\[ m = 4 \to -\frac{1}{4} \qquad m = -\frac{2}{3} \to \frac{3}{2} \qquad m = \frac{1}{5} \to -5 \]
Find the equation of the line perpendicular to \( y = 2x + 1 \) that passes through \( (4, 3) \).

Gradient of \( y = 2x+1 \) is 2. Perpendicular gradient: \( m = -\frac{1}{2} \).

\[ y - 3 = -\frac{1}{2}(x-4) \implies y = -\frac{1}{2}x + 5 \]
Find the equation of the perpendicular bisector of the segment joining \( A(2, 1) \) and \( B(6, 5) \).

Midpoint: \( M = (4, 3) \). Gradient of \( AB \): \( m = \frac{5-1}{6-2} = 1 \). Perpendicular gradient: \( -1 \).

\[ y - 3 = -(x - 4) \implies y = -x + 7 \]

 Key Takeaways

  • Perpendicular lines: \( m_1 \times m_2 = -1 \). The gradient of the perpendicular is the negative reciprocal: \( m_\perp = -\dfrac{1}{m} \).
  • To find the perpendicular gradient: flip the fraction and change the sign.
  • Perpendicular bisector of a segment: passes through the midpoint at the perpendicular gradient.
  • To verify perpendicularity: multiply the two gradients — if the result is \(-1\), the lines are perpendicular.