# Distance from a Point to a Line

The distance from a point to a straight line is the length of a line segment drawn from the point that forms a perpendicular angle with the straight line.

Find the distance from Point P = (2, −1) to the line r ≡ 3 x + 4 y = 0.

### Distance to the Origin

## Examples

Find the distance to the origin of the line r ≡ 3x − 4y − 25 = 0.

A straight line is parallel to another which has an equation of r ≡ 5x + 8y − 12 = 0, and is 6 units from the origin. What is the equation of this line?

A straight line is perpendicular to another that has an equation of r ≡ 5x − 7y + 12 = 0 and is 4 units away from the origin. What is the equation of this line?

### Distance between Parallel Lines

To find the distance between two parallel lines, take any point P from one of the lines and calculate the direct distance to the other.

#### Example

Find the distance between the lines: r ≡ 3 x − 4 y + 4 = 0 and s ≡ 9 x − 12 y − 4 = 0.

The distance between two parallel lines also can be expressed as follows:

#### Example

Calculate the distance between the following lines: