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4.5 – Parallel and Perpendicular Lines

Key Terms

  • Perpendicular – Crossing at a right angle.
    • The symbol \perp means “is perpendicular to.”
    • Slopes of perpendicular lines are negative reciprocals of one another: \frac{4}{1} and \frac{-1}{4}
    • If you multiply the slopes of two perpendicular lines, the product of their slopes is -1.
      • Example: \frac{4}{1}\cdot\frac{-1}{4}=-1
  • Parallel Lines – Have the same slope, but different y-intercepts

 

Notes

  • Parallel Lines
    • If two lines are parallel, their slopes are equal (the same).
      • Parallel Examples: The three lines below all have the same slope: 4
        • y = 4x + 2
        • y = 4x – 3
        • y = 4x + 15

 

  • Perpendicular Lines
    • If two slopes are perpendicular, they have negative reciprocal slopes.
      • They may possibly have different y-intercepts.
      • You MUST know the point where the 2 lines intersect.
    • Horizontal and Vertical Lines
      • If the slope of a line is 0 (horizontal), the slope of perpendicular line is undefined (vertical).
      • If the slope of a line is undefined (vertical), the slope of perpendicular line is 0 (horizontal).

Alg1B 4.5 Perpendicular Zero Undef Red Green

 

  • Perpendicular Example: Intersection Point = (-2, -1), Slope = 4, Negative Reciprocal Slope = \frac{-1}{4}
    • Use Point-Slope Form: y-y_1=m(x-x_1)
      • Step 1: Substitute the known point and slope: y-(-1)=4(x-(-2))
      • Step 2: Simplify: y+1=4(x+2)
      • Step 3: Change to Slope-Intercept Form
        • Distribute the 4: y+1=4x+8
        • Subtract 1 on both sides: y=4x+7
      • Step 4: Now, substitute the known point and the negative reciprocal slope: y-(-1)=\frac{-1}{4}(x-(-2))
      • Step 5: Simplify: y+1=\frac{-1}{4}(x+2)
      • Step 6: Change to Slope-Intercept Form
        • Distribute the \frac{-1}{4}: y+1=\frac{-1x}{4}+\frac{-2x}{4}
        • Simplify: y+1=\frac{-x}{4}+\frac{-x}{2}

 


More Examples

  • Ex 1. The lines below are parallel. If the slope of the green line is -3, what is the slope of the red line?

Alg1B 4.5 Parallel Lines Red Green

Answer: Since parallel lines have the same slope, the red line has a slope of -3 also!

  • Ex 2. The lines below are perpendicular. If the slope of the green line is \frac{3}{2}, what is the slope of the red line?

Alg1B 4.5 Perpendicular Lines Red Green

Answer: Since perpendicular lines have negative reciprocal slopes, the red line has a slope of \frac{-2}{3}.

 

  • Ex 3. Two lines are perpendicular. If one line has a slope of \frac{1}{21}, what is the slope of the other line?
    • Answer: -21

 


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