Distance between two parallel lines

The distance between two parallel lines in the plane is the perpendicular distance from a point on one line to the other line.

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 intercepts 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 intercept 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

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