Dividing a Line Segment into Equal Parts

Date: 11/22/2004 at 08:23:05
From: Iain
Subject: Determining coordinates of equal parts of a line segment

I have two endpoints of a line segment with coordinates A(2, 7) and
B(-4, -2).  I am looking for the coordinates of the points that divide
AB into 3 equal parts.

If drawn on a graph, the points that divide AB are shown clearly.  But
what if the coordinates of AB do not allow such convenient results? 

Date: 11/22/2004 at 09:41:20
From: Doctor Barrus
Subject: Re: Determining coordinates of equal parts of a line segment

Hi, Iain!

Let's look at a picture of a line segment:

        O A(2, 7)
       /
      /
     /
    /
   /
  O B(-4, -2)

Say we want to split segment AB into 3 equal parts, like this:

        O A(2, 7)
       /
      . D(?, ?)
     /
    . C(?, ?)
   /
  O B(-4, -2)

If I understand you correctly, you want to find out the coordinates 
of points C and D, right?  Well, in order to help you understand my 
answer, I'm going to add a little bit more to my drawing.  First I'll 
draw a vertical line through B and a horizontal line through A, which 
will form a triangle:

        O A(2, 7)
       /|
    D . |
     /  |
   C.   |
   /    |
  O-----O E(2,-2)
  B(-4, -2)

Notice that since E is directly below A, its x-coordinate will be 2, 
the same as A's.  Since E is directly to the right of B, its y-
coordinate will be -2, the same as y's.

Now I'm going to mark points on segment BE directly below points C and 
D, and I'm going to mark points on segment AE directly to the right of 
C and D:

        O A
       /|
    D . . H
     /  |
   C.   . I
   /    |
  O-.-.-O E
  B F G

Now the x-coordinate of point C is the same as the x-coordinate of 
point F, right?  So I'm going to try to find the x-coordinate of point 
F, and when I do, the answer will also be the x-coordinate of point C.

Let's just look at that bottom side of the triangle, segment BE:

  -4                      2
  O-------O-------O-------O
  B       F       G       E

The x-coordinate of B is -4, and the x-coordinate of E is 2.  So the 
distance between B and E is 6, because 2 - (-4) = 6.

  -4                      2
  O-------O-------O-------O
  B       F       G       E
   <--------- 6 --------->

Now F is 1/3 of the way from B to E.  Since the total distance from B 
to E is 6, the distance from B to F is (1/3)*6 = 2 (Here * means 
multiplication).  So F is 2 units away from B, and G is 2 units away 
from F.  So we get the picture

  -4      -2      0       2
  O-------O-------O-------O
  B       F       G       E
   <- 2 -> <- 2 -> <- 2 ->

So the x-coordinate of F is -2, and the x-coordinate of G is 0.  If 
we look back the triangle we drew, we see that C also has to have x-
coordinate -2, and D has to have x-coordinate 0.

        O A
       /|
    D . . H
     /  |
   C.   . I
   /    |
  O-.-.-O E
  B F G

Now look at the segment AE with its y-coordinates:

   7 O A 
     |
     |
     O H
     |
     |
     O I
     |
     |
  -2 O E

Can you find out what the y-coordinates for H and I should be?  The 
answers should tell you what the y-coordinates for D and C should be.

I hope this has helped.  If you'd like a little more explanation, 
please write us back with your questions.  Good luck!

- Doctor Barrus, The Math Forum
  http://mathforum.org/dr.math/