3.3: Parallel and Perpendicular Lines | Algebra 1 Honors

How it works

Distinct parallel nonvertical lines have equal slopes and different intercepts. Perpendicular nonvertical lines have slopes whose product is -1. Horizontal and vertical lines are perpendicular, a special case not covered by the product rule because a vertical slope is undefined.

m_1=m_2\ \text{(parallel)},\qquad m_1m_2=-1\ \text{(perpendicular)}

Worked examples

Example 1

Find the line parallel to $\displaystyle y=2x+1$ through $\displaystyle (3,4)$ .

Use the same slope: $\displaystyle y-4=2(x-3)$ , giving $\displaystyle y=2x-2$ .

Example 2

Find the line perpendicular to $\displaystyle y=-\frac12x+3$ through $\displaystyle (0,-1)$ .

The perpendicular slope is $\displaystyle 2$ , since $\displaystyle (-1/2)\cdot2=-1$ . The line is $\displaystyle y=2x-1$ .

Problems

Problem 3.3.1

Are $\displaystyle y=3x+2$ and $\displaystyle y=3x-5$ parallel, perpendicular, or neither?

Show worked solution

They have equal slopes and different intercepts, so they are parallel.

Problem 3.3.2

Find a line perpendicular to $\displaystyle y=4x+7$ through $\displaystyle (0,2)$ .

Show worked solution

The negative reciprocal slope is $\displaystyle -1/4$ . The line is $\displaystyle y=-x/4+2$ .

Problem 3.3.3

What line through $\displaystyle (2,5)$ is perpendicular to $\displaystyle x=2$ ?

Show worked solution

$\displaystyle x=2$ is vertical. The perpendicular line is horizontal through the point: $\displaystyle y=5$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 3.3.1

Problem 3.3.2

Problem 3.3.3

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