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# Math Problem - Finding X2 & Y2 coordinate

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I have a problem. I am trying to find the coordinates of a point along a line at a given distance from a given point on the same line. What I mean is I have X1, X2, slope and distance, and I want to find the point (X2,Y2) that would be that distance from X1, X2 at that slope. I tried combining the distance formula: Sqr((X1 - X2) ^ 2 + (Y1 - Y2) ^ 2) And the slope formula: (Y2 - Y1) / (X2-X1) = m to (Y2-Y1) = (X2-X1)m by substituting the Y''s To get this: (X2-X1)2 + ((X2-X1)m)2 = distance2 The problem is that I have the value of M(slope) and the distance, as well as X1, X2. So If I was manually doing the math, I would have enough variables to figure it out. However, I am trying to throw this into VBasic, so I will need to have the X2 (the unknown) separated from the the rest of the formula. Is there a way for me to do this? Also, I am aware that there will be two points at a given distance from the original point depending on the direction of the line, but I am not worried about that yet.

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X2^2 - 2*X2*X1 + X1^2 - distance^2/(1+m) = 0

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The answer is much simpler than what you are trying to do. You just need to remember about Thales law.
Reminder:
the equation of a line is expressed under the form: y = ax + b, where a is the slope.
the distance between two points A(x1,y1) and B(x2,y2) is: d=sqrt( (x2-x1)^2 + (y2-y1)^2)
Thales law: if several parallels intersect two straightlines, they''ll determine proportional segments.

OK, let''s start to find the equation giving us the coordinates of point C(x3,y3) knowing distance d and the point A(x1,y1):

the elementary distance when you increment your x variable by 1 is equal to:
delem = sqrt (1 + a^2) since (x2-x1) = 1 and (y2-y1) = slope.

now using Thales law, you can say that
delem / d = 1 / (x3 - x1) or d / delem = (x3 - x1) / 1 which is strictly equivalent.

thus you find that x3 = x1 + (d / delem)

To derive the y3 value, you just use your line equation: y3 = ax3 + b.

Hope that helps.
Ghostly yours,
Red.

What is delem?

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Sorry, I figured out what you were saying.
So the final formula would be
x3 = x1 + (d / sqrt (1 + a^2))

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quote:
Original post by kevinheeney
What is delem?

d-elem = distance elementary = elementary distance

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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