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Learn Index · Functions and Relations Grade 9–10

Parallel and perpendicular lines

Parallel lines have equal slopes; perpendicular slopes are negative reciprocals.

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Parallel lines

Parallel lines never meet, because they rise at exactly the same rate.

Rule: parallel lines have equal slopes.

  • y = 3x + 1 and y = 3x − 5 are parallel (both have m = 3).
  • They differ only in their y-intercepts.

Perpendicular lines

Perpendicular lines meet at a right angle (90°).

Rule: their slopes are negative reciprocals. Equivalently, the product of slopes is −1.

Slope Negative reciprocal
2 −1/2
−3/4 4/3
1 −1
−5 1/5

Check: 2 × (−1/2) = −1 ✓

Special cases

  • A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope).
  • All horizontal lines are parallel to each other; so are all vertical lines.

Identifying the relationship

  1. Write each equation in slope-intercept form.
  2. Compare slopes.

Example: 2x + y = 7 and x − 2y = 4

  • y = −2x + 7 → m = −2
  • y = ½x − 2 → m = ½
  • −2 × ½ = −1 → perpendicular

Writing equations

Parallel line through a point: Find the line parallel to y = 4x − 1 through (2, 3).

  • m = 4
  • 3 = 4(2) + b → b = −5
  • y = 4x − 5

Perpendicular line through a point: Find the line perpendicular to y = ⅔x + 2 through (4, 1).

  • m = −3/2
  • 1 = −3/2(4) + b → 1 = −6 + b → b = 7
  • y = −3/2 x + 7

Geometry application

Show that A(0, 0), B(4, 2), C(3, 4), D(−1, 2) form a rectangle:

  • AB slope = 2/4 = ½; DC slope = 2/4 = ½ → parallel
  • AD slope = 2/(−1) = −2; BC slope = 2/(−1) = −2 → parallel
  • ½ × (−2) = −1 → adjacent sides perpendicular → rectangle ✓

Worked examples

Example 1: Are y = −5x + 2 and y = −5x + 9 parallel? Yes.

Example 2: Slope perpendicular to y = 7x? −1/7.

Example 3: Are y = 2x and y = −2x perpendicular? 2 × (−2) = −4 ≠ −1 → no.

Common mistakes

  • Only flipping, or only changing the sign, instead of both.
  • Comparing equations not in slope-intercept form.
  • Thinking same y-intercept means parallel.

Try it yourself

  1. Slope parallel to y = −⅓x + 4?
  2. Slope perpendicular to y = −⅓x + 4?
  3. Are 3x + y = 5 and x − 3y = 9 parallel, perpendicular or neither?
  4. Line parallel to y = 2x + 1 through (0, −6)?
  5. Line perpendicular to y = x through (3, 3)?

Answers: 1) −⅓ 2) 3 3) perpendicular 4) y = 2x − 6 5) y = −x + 6

Quick recap

  • Parallel lines: equal slopes.
  • Perpendicular lines: slopes are negative reciprocals (product −1).
  • Use the point and slope to write new equations.