# Precalculus : Find the Distance Between Two Parallel Lines

## Example Questions

### Example Question #1 : Find The Distance Between Two Parallel Lines

Find the distance  between the two lines.

Explanation:

Since the slope of the two lines are equivalent, we know that the lines are parallel. Therefore, they are separated by a constant distance. We can then find the distance between the two lines by using the formula for the distance from a point to a nonvertical line:

First, we need to take one of the line and convert it to standard form.

where

Now we can substitute A, B, and C into our distance equation along with a point, , from the other line. We can pick any point we want, as long as it is on line . Just plug in a number for x, and solve for y. I will use the y-intercept, where x = 0, because it is easy to calculate:

Now we have a point, , that is on the line . So let's plug our values for :

### Example Question #2 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in 2 into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #3 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the second equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #4 : Find The Distance Between Two Parallel Lines

How far apart are the lines and ?

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #5 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the second equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #6 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #7 : Find The Distance Between Two Parallel Lines

Find the distance between the lines and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.