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Finding the Other End of a Line Segment

Date: 07/25/2002 at 20:29:26
From: Dawn Langseth
Subject: Finding a point

I know a point location and I know the slope of the line. I also know 
the distance. How do I find the point at the other end of the line?

Date: 07/25/2002 at 23:11:04
From: Doctor Peterson
Subject: Re: Finding a point

Hi, Dawn.

I would use vectors, though you don't need to know anything about them 
to do this. Suppose the slope is 3/4. Then one vector (directed 
segment) in this direction is (4,3) - 4 across and 3 up. If you draw 
a point 4 across and 3 up from the starting point, it will be in the 
right direction, but probably the wrong distance.

So we find the length of our vector, 

  sqrt(3^2 + 4^2) = sqrt(25) = 5, 

(which I chose to make the numbers easy), and divide by it. Our new 
vector, whose length will be 1, is (4/5,3/5). 

Now to get a vector of the desired length (say, 3) we can multiply by
that and get (12/5,9/5). Just add that vector to your starting point
(that is, add 12/5 to the x coordinate and 9/5 to y) and you have a
point that is both in the right direction and at the right distance.
Check the distance and see.

I should mention that there are actually TWO points along the given 
line and at the right distance. I've chosen one direction; you can 
subtract my vector rather than adding it to get the other possibility.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Coordinate Plane Geometry
High School Linear Algebra

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