Finding Points on a Line in Slope-Intercept Form

Date: 07/29/2005 at 13:24:37
From: Ben
Subject: How to find the other two points on a line

Hi, Dr. Math:

I have a line y = kx + b, and I know point A(x1, x2) is on the line. 
If point B is also on the line, and the distance between A and B is d, 
what are the point(s) B?

Thanks,
Ben

Date: 07/29/2005 at 15:17:16
From: Doctor Jaffee
Subject: Re: How to find the other two points on a line

Hi Ben,

Here's how I solved this problem.  I drew a line with the point A(x1, 
x2) on it and a point B(z1,z2).  Actually, there are two possible 
locations for point B.  One is to the right and above (if k is a 
positive number) and the other is to the left and below.  If k is 
negative, then B could be to the right and below A or to the left and 
above A.  Of course, if k is 0, then the problem is easy to solve.  If 
k = 0, the line is horizontal, so all that you have to do is add d to 
x1 to get z1 and z2 stays the same.  Then, subtract d from x1 to find 
the other point.

But, if the line is not horizontal, you have to work a little harder.  
Assume that the horizontal distance from A to B is p and the vertical 
distance from A to B is q.  Then, the slope of the line is q/p = k.  
According to the Pythagorean Theorem p^2 + q^2 = d^2.  Now, the values 
for k and d are constants, so solve this system of equations for q and 
p and you can then easily determine both locations for B.

Give it a try and you have any difficulties or you have questions 
about my suggestions, write back and I'll try to help you some more.  

Good Luck,
- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/29/2005 at 15:25:25
From: Ben
Subject: Thank you (How to find the other two points on a line)

Thanks Doctor Jaffee! That helps a lot!