Slopes of Parallel and Perpendicular Lines
Connect the geometric properties of parallel and perpendicular lines to their slopes in the coordinate plane. Write equations of lines parallel or perpendicular to a given line through a given point.
A lesson of Geometry, in Parallel and Perpendicular Lines, about 10 minutes.
By the end of this lesson, you will be able to determine if lines are parallel, perpendicular, or neither using their slopes, write equations of lines parallel or perpendicular to a given line through a given point.
The Big Idea
In the coordinate plane, the geometric relationship between two lines — parallel, perpendicular, or neither — is encoded entirely in their slopes. This bridges Algebra 1 (linear equations) and Geometry (parallel/perpendicular theorems) in a single tool.
Slope Review
The slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$$
Slope measures steepness and direction. Positive slope: rises left to right. Negative slope: falls left to right. Zero slope: horizontal. Undefined slope: vertical.
Parallel Lines — Equal Slopes
Two non-vertical lines are parallel if and only if they have equal slopes:
$$m_1 = m_2$$
Vertical lines are parallel to each other (both have undefined slope).
Perpendicular Lines — Negative Reciprocal Slopes
Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals:
$$m_1 \cdot m_2 = -1 \quad \Leftrightarrow \quad m_2 = -\frac{1}{m_1}$$
A horizontal line ($m = 0$) is perpendicular to a vertical line (undefined slope).
Finding the negative reciprocal: flip the fraction and change the sign.
- If $m = 2$, the negative reciprocal is $-\dfrac{1}{2}$
- If $m = -\dfrac{3}{4}$, the negative reciprocal is $\dfrac{4}{3}$
Example 1 — Classifying Line Pairs
Classify each pair:
(a) $y = 3x + 1$ and $y = 3x - 7$: slopes both = 3 → parallel
(b) $y = 2x + 5$ and $y = -\dfrac{1}{2}x + 3$: slopes $2$ and $-\dfrac{1}{2}$, product = $-1$ → perpendicular
(c) $y = 4x + 1$ and $y = -4x + 1$: slopes $4$ and $-4$, product = $-16 \neq -1$ → neither
Example 2 — Writing a Parallel Line
Write the equation of the line parallel to $y = -2x + 3$ passing through $(1, 5)$.
Parallel → same slope: $m = -2$
$$y - 5 = -2(x - 1) \implies y = -2x + 7$$