Parallel and Perpendicular Lines

By the end of this lesson, you will be able to tell from two equations whether their lines are parallel, perpendicular or neither, and write a line through a given point that is parallel or perpendicular to another.

Two Rules, One Idea

Parallel lines have equal slopes.

$$m_1 = m_2$$

Slope is direction, so lines pointing the same way never meet. Their intercepts must differ, though: same slope AND same intercept is not two parallel lines, it is one line written twice.

Perpendicular lines have slopes that multiply to $-1$.

$$m_1 \cdot m_2 = -1$$

Which is easier to use as: flip it and change the sign.

$$m = \frac{2}{3} \longrightarrow m_\perp = -\frac{3}{2}$$

$$m = 4 \longrightarrow m_\perp = -\frac{1}{4}$$

$$m = -\frac{1}{5} \longrightarrow m_\perp = 5$$

Flip, then negate. Both steps, every time. Only flipping gives a parallel-ish line that is not perpendicular; only negating does the same.

Why the Negative Reciprocal

It is worth one paragraph, because the rule looks arbitrary and is not.

A slope of $\tfrac{2}{3}$ is the instruction "right 3, up 2". Rotate that instruction by ninety degrees and right becomes up, while up becomes left. So "right 3, up 2" turns into "up 3, left 2", which as a slope is $\tfrac{3}{-2}$, which is $-\tfrac32$.

The rise and run swap because the rotation swaps the axes, and one sign flips because one of the two directions reverses. That is the whole derivation, and it means the rule can be rebuilt from a picture rather than remembered.

Example 1: Deciding from Two Equations

Here is the whole method on one pair, worked all the way through. $$y = 3x + 1 \qquad 6x + 2y = 8$$

• The first slope is visible. What is it? 3
• Rearrange the second into slope-intercept form y = -3x + 4
• What is the second slope? -3
• Multiply the two slopes together -9
• Parallel, perpendicular, or neither? neither

Sort It: Parallel, Perpendicular or Neither

Each pair of slopes below. Which relationship do they have?

Worked Example 1: Finding the Partner Slope

Writing the Line

Both jobs are the same three moves: get the given slope, adjust it, then use point-slope with the point you were given.

Write the line through $(2, 5)$ parallel to $y = 4x - 1$.

Slope $4$, unchanged. $y - 5 = 4(x - 2)$, so $y = 4x - 3$.

Write the line through $(2, 5)$ perpendicular to $y = 4x - 1$.

Slope $-\tfrac14$. $y - 5 = -\tfrac14(x - 2)$, so $y = -\tfrac14 x + \tfrac{11}{2}$.

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