Understanding Perpendicular Lines and Equations

Apr 29, 2025

Lecture on Equations of Lines

Key Concepts

  • Equation of a Line: Typically presented in slope-intercept form as ( y = mx + b )
    • ( m ) = Slope of the line
    • ( b ) = Y-intercept of the line

Perpendicular Lines

  • Definition: Perpendicular lines have slopes that are opposite reciprocals.
    • For a slope ( m ), the perpendicular slope is (-1/m).
    • If the original slope is negative, the perpendicular slope will be positive and vice versa.

Example

  1. Given Slope: (-2)

    • Perpendicular slope: (1/2)
  2. Forming a New Equation:

    • Start with ( y = \frac{1}{2}x + b )
  3. Finding the Y-intercept ((b)):

    • Use a point ((x, y)) to solve for ( b ).
    • Example Point: ((-3, -2))
  4. Calculation:

    • Plug in values:
      • ( -2 = \frac{1}{2}(-3) + b )
    • Simplify:
      • ( -2 = -\frac{3}{2} + b )
    • Add ( \frac{3}{2} ) to both sides:
      • ( -2 + \frac{3}{2} = b )
      • Simplify ( -2 + 1.5 = -0.5 )
    • Therefore, ( b = -\frac{1}{2} )
  5. Complete Equation:

    • The new equation of the line: ( y = \frac{1}{2}x - \frac{1}{2} )

Summary

  • To find the equation of a line perpendicular to a given line:
    • Determine the reciprocal and opposite of the original slope for the perpendicular slope.
    • Use a given point to solve for the new y-intercept ( b ).
    • Form the equation with the new slope and y-intercept.