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# 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 $6x+8y=6$ and $6x+8y=26$ .

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.

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

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

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.

b. True or false: Parallel lines intersect.

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

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

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