Distance between two parallel lines

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

Formula and proof

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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

 
 

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

 

This distance can be found by first solving the linear systems

 

and

 

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

 

and

 

The distance between the points is

 

which reduces to

 

When the lines are given by

 
 

the distance between them can be expressed as

 

See also

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References

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  • Abstand In: Schülerduden – Mathematik II. Bibliographisches Institut & F. A. Brockhaus, 2004, ISBN 3-411-04275-3, pp. 17-19 (German)
  • Hardt Krämer, Rolf Höwelmann, Ingo Klemisch: Analytische Geometrie und Lineare Akgebra. Diesterweg, 1988, ISBN 3-425-05301-9, p. 298 (German)
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