Parallel & Perpendicular Lines

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Determine whether two lines are parallel, perpendicular, or neither.
  • Write equations of lines that are parallel or perpendicular to a given line.
  • Use slope relationships confidently when building new equations.

Key Ideas

Understanding slope relationships is the key to quickly identifying how two lines behave.

Parallel Lines

  • Have the same slope
  • Have different y-intercepts
  • Never intersect

Example:
\(y = 3x + 1\) and \(y = 3x - 5\) are parallel because both have \(m = 3\).

Two parallel lines with the same slope.

Perpendicular Lines

  • Slopes are opposite reciprocals:

\[ m_2 = -\frac{1}{m_1} \]

Example:
If one slope is \(2\), the perpendicular slope is \(-\frac{1}{2}\).

Vertical and Horizontal

  • Vertical line: \(x = c\)
  • Horizontal line: \(y = k\)

These two are always perpendicular.

Two perpendicular lines with opposite reciprocal slopes.

Common Problem Types

1. Determine if lines are parallel, perpendicular, or neither

Compare slopes:

  • same slope → parallel
  • negative reciprocals → perpendicular
  • anything else → neither

2. Write a line parallel to a given line

Use the same slope, different intercept, and pass it through a given point.

3. Write a perpendicular line through a point

Use the opposite reciprocal slope, substitute the point, and create the new equation.

4. Rewrite from Standard Form if needed

Convert \(Ax + By = C\) to slope-intercept to identify the slope quickly.

Strategies

  • Always extract slope first—even if the equation is not in slope-intercept form.
  • When writing a new line, use point-slope form to avoid early algebra mistakes.
  • Keep perpendicular slopes straight:
    • Flip the fraction
    • Change the sign
  • Check your final equation by confirming it passes through the given point.

Worked Examples

Example 1 — Parallel through a point

Line:
\[ y = 4x + 2 \]

Find a parallel line through \((1, -3)\).

Parallel slope = 4.

Use point-slope:

\[ y + 3 = 4(x - 1) \]

(You may convert if desired.)

Example 2 — Perpendicular through a point

Line:
\[ y = -3x + 5 \]

Given slope = \(-3\).
Perpendicular slope:

\[ m = \frac{1}{3} \]

Through point \((2,1)\):

\[ y - 1 = \frac{1}{3}(x - 2) \]

WarningCommon Mistakes
  • Flipping a slope for perpendicular lines but forgetting to change the sign.
  • Thinking perpendicular slopes multiply to +1 instead of -1.
  • Mixing up vertical and horizontal lines.
  • Using the intercept from the original line instead of substituting the new point.

Practice Problems

  1. Write the equation of a line parallel to \(y = 5x - 1\) through \((2, 4)\).
  2. Write the equation of a line perpendicular to
    \(y = -\frac{1}{2}x + 3\) through \((0, -1)\).
  3. Determine whether the lines are parallel, perpendicular, or neither:
    • \(y = 2x + 3\)
    • \(4x - 2y = 10\)
  4. Find the equation of a line perpendicular to \(x = 3\) through \((2, 5)\).
  5. Write a parallel line to \(3x + y = 7\) through \((0, -2)\).

1.
Parallel slope = 5
\[ y - 4 = 5(x - 2) \]
Convert:
\[ y = 5x - 6 \]

2.
Slope given = \(-\frac{1}{2}\)
Perpendicular slope = \(2\)
\[ y + 1 = 2(x - 0) \Rightarrow y = 2x - 1 \]

3.
Rewrite second:
\[ -2y = -4x + 10 \Rightarrow y = 2x - 5 \]
Same slope → parallel.

4.
\(x = 3\) is vertical.
Perpendicular line = horizontal → \(y = 5\).

5.
Rewrite original for slope:
\[ y = -3x + 7 \]
Parallel slope = \(-3\)
Through \((0,-2)\):
\[ y + 2 = -3(x - 0) \Rightarrow y = -3x - 2 \]

Summary

  • Parallel lines share the same slope.
  • Perpendicular lines have opposite reciprocal slopes.
  • Vertical and horizontal lines are always perpendicular.
  • Use point-slope form when writing equations from a given point.
  • “Parallel = same slope; Perpendicular = flip and negate.”
  • If the equation isn’t in \(y = mx + b\) form, rewrite it first.
  • When unsure, pick one point and check that it satisfies your final equation.