Find the coordinates of the points that divide the segment A(-6, -3) and B(3, 1) into three equal parts.

Answer 1

let P(x,y) divide AB into 1:3

so for the y:
(y+3)/(1-y) = 1/3
3y + 9 = 1 - y
4y = -8
y = -2

for the x:
(x+6)/(3-x) = 1/3
3x + 18 = 3-x
4x = -15
x = -15/4 , so P is (-15/4, -2)

check so far:
http://www.wolframalpha.com/input/?i=plot+%7B+(-6,-3),+(3,1)+,+(-15%2F4,-2)%7D
notice the 3 points all fall on the same line and AP : PQ looks like 1:3

let Q(x,y) divide AB in the ration of 2 : 3

after you find Q, go back to the Wolfram page and edit in your new point

Answer 2

let's back up a bit here.
I set up a ratio of 1:3, thus splitting the line into 4 parts
SHOULD HAVE BEEN RATION 1:2

let P(x,y) divide AB into 1:2

so for the y:
(y+3)/(1-y) = 1/2
2y + 6 = 1 - y
3y = -5
y = -5/3

for the x:
(x+6)/(3-x) = 1/2
2x + 12 = 3-x
3x = -9
x = - 3 , so P is (-3, -5/3)

Revised Wolfram"
http://www.wolframalpha.com/input/?i=plot+%7B+(-6,-3),+(3,1)+,+(-3,+-5%2F3)%7D

That's better.

Answer 3

The change in coordinates from A(-6, -3) to B(3, 1) is

(3-(-6),1-(-3)) = (9,4)
1/3 of the difference is thus (3,4/3)

(-6,-3) + (3,4/3) = (-3,-5/3)
(-3,-5/3) + (3,4/3) = (0,-1/3)
(0,-1/3) + (3,4/3) = (3,1)

The first two points divide AB into 3 equal parts.