# Distance between two parallel lines

The distance between two parallel lines in the plane is the minimum distance between any two points.

## Formula and proof

Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the equations of two non-vertical parallel lines

${\displaystyle y=mx+b_{1}\,}$
${\displaystyle y=mx+b_{2}\,,}$

the distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line

${\displaystyle y=-x/m\,.}$

This distance can be found by first solving the linear systems

${\displaystyle {\begin{cases}y=mx+b_{1}\\y=-x/m\,,\end{cases}}}$

and

${\displaystyle {\begin{cases}y=mx+b_{2}\\y=-x/m\,,\end{cases}}}$

to get the coordinates of the intersection points. The solutions to the linear systems are the points

${\displaystyle \left(x_{1},y_{1}\right)\ =\left({\frac {-b_{1}m}{m^{2}+1}},{\frac {b_{1}}{m^{2}+1}}\right)\,,}$

and

${\displaystyle \left(x_{2},y_{2}\right)\ =\left({\frac {-b_{2}m}{m^{2}+1}},{\frac {b_{2}}{m^{2}+1}}\right)\,.}$

The distance between the points is

${\displaystyle d={\sqrt {\left({\frac {b_{1}m-b_{2}m}{m^{2}+1}}\right)^{2}+\left({\frac {b_{2}-b_{1}}{m^{2}+1}}\right)^{2}}}\,,}$

which reduces to

${\displaystyle d={\frac {|b_{2}-b_{1}|}{\sqrt {m^{2}+1}}}\,.}$

When the lines are given by

${\displaystyle ax+by+c_{1}=0\,}$
${\displaystyle ax+by+c_{2}=0,\,}$

the distance between them can be expressed as

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