Theorem 1.12: Parallel lines stay the same distance apart. If the equations of the lines are

then the perpendicular distance from any point on one line to the other line is

If the lines are vertical, then their equations are

x = a1   and   x = a2

and the distance between the lines is

|a1 - a2|

Proof: If   (x1, y1)   is a point on the line whose equation is   y = mx + b1,   then by Theorem 1.3,

y1 - mx1 = b1

and the result follows from Theorem 1.11.

If the lines are vertical then a point on   x = a1   would have the form   (a1, y1).   The line which is parallel to the first vertical line would also be vertical, and the line from   (a1, y)   perpendicular to the other vertical line would be horizontal, and its equation would be

y = y1

The point of intersection between this line and the vertical line

x = a2

would be

(a2, y1)

The distance between   (a1, y1)   and   (a2, y1)   would be

|a1 - a2|

by a simple application of the distance formula.

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