DWG NO. 03.3 — Lesson 3 of 4

Slope, and Parallel vs. Perpendicular Lines

Unit 3: Parallel and Perpendicular Lines · ~15–30 min

Objective

Calculate the slope of a line from two points, and use slope to determine whether two lines are parallel, perpendicular, or neither.

The slope formula

Slope measures steepness — how much a line rises or falls for each step it takes sideways. For two points (x₁, y₁) and (x₂, y₂) on a line:

slope m = (y₂ − y₁) ÷ (x₂ − x₁) = rise ÷ run

A positive slope rises left to right; a negative slope falls. A horizontal line has slope 0. A vertical line has undefined slope — the run is 0, and division by zero isn't defined.

Slope and parallel vs. perpendicular lines

Parallel lines have the same slope. If line a has slope m and line b has slope m, then a ∥ b (as long as they aren't the same line).
Perpendicular lines have slopes that are negative reciprocals of each other — flip the fraction and change the sign. If line a has slope m, a line perpendicular to it has slope −1/m. Equivalently, the product of the two slopes is −1.
A horizontal line (slope 0) and a vertical line (undefined slope) are always perpendicular to each other, even though the "negative reciprocal" rule can't literally be applied to 0.

Line a has slope 2/3. Line b, perpendicular to a, has slope −3/2 — the negative reciprocal.

Worked Example 1 · Finding slope from two points

ProblemFind the slope of the line through (0, 2) and (3, 4).

1.Apply the formula: m = (4 − 2) ÷ (3 − 0).

2.Simplify: m = 2 ÷ 3.

slope = 2/3

Worked Example 2 · Parallel, perpendicular, or neither?

ProblemLine a passes through (0, 2) and (3, 4). Line b passes through (1, 5) and (3, 2). Are lines a and b parallel, perpendicular, or neither?

1.Slope of line a (from Example 1): 2/3.

2.Slope of line b: m = (2 − 5) ÷ (3 − 1) = −3 ÷ 2 = −3/2.

3.Multiply the slopes: (2/3)(−3/2) = −1. A product of −1 means the slopes are negative reciprocals.

Lines a and b are perpendicular

Guided practice

1.Find the slope of the line through (−2, 1) and (2, 9).
Show answer
m = (9 − 1) ÷ (2 − (−2)) = 8 ÷ 4 = 2.
2.Line p has slope 5/4. What slope must a line have to be parallel to p? To be perpendicular to p?
Show answer
Parallel: 5/4 (the same slope). Perpendicular: −4/5 (the negative reciprocal).
3.Line r passes through (0, 0) and (4, 6). Line s passes through (1, 1) and (5, 7). Are r and s parallel, perpendicular, or neither?
Show answer
Parallel. Slope of r = 6/4 = 3/2. Slope of s = (7 − 1) ÷ (5 − 1) = 6/4 = 3/2 — the same slope.
4.A line contains the points (3, 2) and (3, 9). What is its slope, and what does that tell you about a line perpendicular to it?
Show answer
The run is 0, so the slope is undefined — the line is vertical. A line perpendicular to it must be horizontal, with slope 0.
5.Line m has slope −2/5. Line n passes through (0, 3) and (5, 1). Are m and n parallel, perpendicular, or neither?
Show answer
Parallel. Slope of n = (1 − 3) ÷ (5 − 0) = −2/5, the same as line m’s slope.

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