Distance between Parallel Lines

Parallel lines are coplanar lines that never intersect. We see parallel lines every day, whether they're power lines strung overhead or underfoot like a crosswalk. Here is an example of parallel lines on a graph:

Distance between two parallel lines

The shortest distance between parallel lines is the distance between two points, one on each line, that forms a perpendicular line to the parallel lines. Perpendicular lines fall at 90 degrees or a right angle. You can draw perpendicular lines between many points, and they will all have the same distance between the parallel lines.

Calculating the distance between parallel lines (standard form)

Let's say you want to find the distance between the parallel lines $6 x + 8 y = 6$ and $6 x + 8 y = 26$ .

These equations are written in the standard form of a linear equation, which is ax+by=c.

For both equations, $a = 6$ and $b = 8$ . Since each c is different, we'll say that $c_{1}$ is the

constant for line 1 and $c_{2}$ is the constant for line 2.

$c_{1} = 6$ and $c_{2} = 26$

The distance formula is:

$d = \frac{| c_{2} - c_{1} |}{\sqrt{a^{2} + b^{2}}}$

Let's substitute the numbers.

$d = \frac{| 26 - 6 |}{\sqrt{6^{2} + 8^{2}}}$

Simplify the numerator.

$d = \frac{| 20 |}{\sqrt{6^{2} + 8^{2}}}$

Simplify the denominator.

$d = \frac{| 20 |}{\sqrt{36 + 64}} = \frac{20}{10}$

Reduce the fraction.

$d = \frac{20}{10} = \frac{2}{1} = 2$

The distance between the two parallel lines is 2.

Practice questions on the distance between parallel lines

a. True or false: The shortest distance between two parallel lines is any two points between them that form a perpendicular line.

Answer: True

b. True or false: Parallel lines intersect.

Answer: False

c. True or false: ax+by=c is written in the standard form of a linear equation.

Answer: True

d. Find the distance between the following parallel lines: $3 x + 4 y = 6$ and $3 x + 4 y = 21$ where $a = 3$ , $b = 4$ , $c_{1} = 6$ , and $c_{2} = 21$ .

Answer: 3

e. Find the distance between the following parallel lines: $5 x + 12 y = 10$ and $5 x + 12 y = 36$ where $a = 5$ , $b = 12$ , $c_{1} = 10$ , and $c_{2} = 36$ .

Answer: 2

Topics related to the Distance between Parallel Lines

Parallel Lines and Slopes

Parallel Lines

Parallel and Perpendicular Lines

Flashcards covering the Distance between Parallel Lines

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Distance between Parallel Lines

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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