Expert: Ahmed Salami - 9/16/2004

Question
Hi,

I am generally ignorant when it comes to coordinate
geometry, so I'll apologize in advance if my question is not
clear. Anyway here it goes...

Assume I have a line with end points (25,30) and (190,125).
I am pretty sure this line has a length of 255.

Now, let's say I want to find the x&y coordinates for a point
along the line which is 80 units from the endpoint (25,30).
Is there an equation which will produce those coordinates?

Thank you!


Answer
Hi Josh,
I know this is terribly late. I had a technical problem with the site. Please forgive me.
Well, to start with, the lenght of the line spanning from (25,30)to(190,125)is not 255 but
sqrt[(190 - 25)^2 + (125 - 30)^2]
= sqrt[(165)^2 + (95)^2]
= sqrt[36250]
= 190.39
Now, to find the coordinates of a point along this line which is 80 units of distance from the (25,30)point there are actually two ways to go about it, but i'll give you the more simpler one.
In coordinate geometry, there is a formula that gives the coordinates of a point on a line when the line is divided in the ratio m:n. If we start from one end and the line is to be divided into a ratio m:n from that end and given that the end points of the line are with coordinates (x1,y1)and(x2,y2), then the coordinates of this point would be;
x = (nx1 + mx2)/(m+n)
y = (ny1 + my2)/(m+n)
So, as regards your question we can set our ratio as 80:110.39. Now,
x1 = 25, x2 = 190, y1 = 30, y2 = 125, m = 80, n = 110.39
Therefore, our coordinates are
x = 110.39(25)+ 80(190)/ 80+110.39
 = 17959.75/190.39
 = 94.33
y = 110.39(30)+ 80(125)/ 80+110.39
 = 13311.7/190.39
 = 69.92

I do hope you get it. Please forgive me about the delay. You can always contact me on anything.
Regards.