Show that the distance between the points (x,y) and the line Ax + By + C = 0 is:

Question

D = (Ax + By + C) / sqrt(A^2 + B^2)

So, would you start out by multiplying sqrt (A^2 + B^2) to both sides and then it would become:
sqrt (A^2 + B^2)D = Ax + By + C

Then square both sides,
D^2(A^2 + B^2) = (Ax + By + C)^2

Then I got lost, could someone direct me through this problem? Did I start out the right way? Thanks!

Answer 1

Just manipulating the equation for D will not constitute a proof that it is correct. You need to derive it somehow.

Try deriving an equation for a line through (x,y) that is perpendicular to the line Ax + By +C = 0. Then compute where the two lines intersect. Call that point (x', y'). Then compute the distance between (x,y) and (x',y').

A proof can be found here:
http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html