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MathChapter 3: Functions and Linear Equations

Notes

Parallel and Perpendicular Lines Notes

Use slope relationships and special vertical/horizontal cases to write and classify lines.

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Parallelism and perpendicularity are slope relationships, not visual guesses. Convert equations to a readable form, compare slopes, and handle vertical and horizontal lines as explicit special cases.

Slope rules

Parallel versus perpendicular

Parallel

Distinct nonvertical parallel lines have the same slope and different intercepts.

Perpendicular

Defined nonzero slopes are negative reciprocals: \(m_1m_2=-1\).

Negative reciprocal relationship
\[m_{\perp}=-\frac{1}{m}\]

Take the reciprocal and change the sign. The rule assumes a defined, nonzero original slope.

Required line relationships

Parallel and perpendicular configurations

Each panel is generated from exact slope, intercept, and line-orientation data.

Sloped parallel linesSloped parallel lines. First line: Line A. Second line: Line B. Their solution classification is identified by their slopes and intercepts.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xyLine ALine B
Sloped parallel lines
Vertical parallel linesDistinct vertical lines x = -2 and x = 3 never meet.-5-4-3-2-112345-5-4-3-2-112345xyLine ALine B
Vertical parallel lines
Horizontal parallel linesDistinct horizontal lines y = -2 and y = 2 have zero slope and never meet.-5-4-3-2-112345-5-4-3-2-112345xyLine ALine B
Horizontal parallel lines
Negative reciprocal slopesNegative reciprocal slopes. First line: Line A. Second line: Line B. They intersect at 0, 0.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xy(0, 0)Line ALine B
Negative reciprocal slopes
Vertical and horizontal linesThe vertical line x = 1 and horizontal line y = -1 meet at a right angle.-5-4-3-2-112345-5-4-3-2-112345xyVertical lineHorizontal line
Vertical and horizontal lines

Write a related line through a point

Worked example

Parallel line

Write the line parallel to \(3x-2y=8\) through \((-1,5)\).

  1. Find original slope

    \(-2y=-3x+8\), so \(m=3/2\).

  2. Keep slope

    A parallel line also has slope \(3/2\).

  3. Use the point

    \(y-5=\frac32(x+1)\).

  4. Optional conversion

    \(2y-10=3x+3\), so \(3x-2y=-13\).

\(y-5=\frac32(x+1)\), or an equivalent form.
Worked example

Perpendicular line

Write the line perpendicular to \(y=-\frac25x+7\) through \((4,-1)\).

  1. Change slope

    The negative reciprocal of \(-2/5\) is \(5/2\).

  2. Use point-slope

    \(y+1=\frac52(x-4)\).

  3. Check

    \((-2/5)(5/2)=-1\).

\(y+1=\frac52(x-4)\).

Use slope to find a missing coordinate

If segment \(AB\) must be parallel or perpendicular to another line, write its slope using the unknown coordinate, set it equal to the required slope, and solve. This turns a geometry relationship into a linear equation.

Mini check

Check your understanding

A line has slope \(-4\). What is the slope of a perpendicular nonvertical line?

  1. \(-4\)
  2. \(4\)
  3. \(-1/4\)
  4. \(1/4\)
Show answer and explanation

Answer: \(1/4\)

The negative reciprocal of \(-4=-4/1\) is \(1/4\), and the product is \(-1\).

Key takeaways

What to remember

  • Parallel nonvertical lines have equal slopes.
  • Perpendicular defined nonzero slopes multiply to \(-1\).
  • The perpendicular operation changes both reciprocal and sign.
  • Vertical lines are parallel to vertical lines and perpendicular to horizontal lines.
  • Use point-slope form after choosing the required slope.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.