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Name: Jeffrey A. H.
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Question:
If one knows the coordinates of one point and of two points defining a line, is there a formula for finding the distance between the line and the point? AND, can that formula be written as:

Distance =

I have been to several web pages and read several equations that define lines and solve for distance, but I can't seem to figure out how they relate to coordinates on a grid...

For example, the followig is given as the definition of a line:

Ax + By + C = 0

... If A and B are points, what is C?

Sorry if I am missing something obvious... Thank you very much for your time.



Replies:
See http://forum.swarthmore.edu/dr.math/problems/albester4.10.98.html

Also, you might enjoy the following site:

http://www.netcomuk.co.uk/~jenolive/index.html

I found these by searching (www.google.com) for the phrase "perpendicular distance".

Tim Mooney


Jeffrey,

There is such a formula, and I can derive it if you understand two things: First, in Ax+By+C=0, A,B,C are NOT points. They are constant values, but the points are combinations of x and y, usually written as (x,y). Second, the shortest distance from a point to a line is along a straight path toward the line: it will be perpendicular to the original line.

I will use a line formula called slope-intercept form: y=mx+b. It works easier for this type of problem. Let y=Mx+B be the line through the two points. Let (u,v) be the coordinates of the point. Any line perpendicular to the original line will have a slope m=-1/M. y=(-1/M)x+b. It must pass through the point (u,v), so the following must be true: v=(-1/M)u + b. Therefore, b=v+u/M. We can say y=(-1/M)x+{v+u/M}. Another format is (y-v)=-(1/M)(x-u).

Dr. Ken Mellendorf
Illinois Central College


Suppose you have two points (X1,Y1) and (X2,Y2) It is possible to rearrange the line between the two points in the form: Y = m*X + b. Do this by "plugging in" Y1=m*X1+b and Y2=m*X2+b and solving the two equations for the two unknowns, 'm' and 'b'. One can show that a line perpendicular to Y = m*X + b has a slope of (-1/m), which gives the perpendicular Y' = (-1/m)*X' + b' Since the coordinates of the the point C = (X', Y') substitution into the the latter equation gives the value of 'b'.

Vince Calder



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