# 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).

• 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?

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?

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?