Find the coordinates of the points that divide the segment A(-6, -3) and B(3, 1) into three equal parts.
To find the coordinates of the points that divide the segment AB into three equal parts, we can use the concept of the midpoint formula.
The midpoint formula states that the coordinates of the point M, which is the midpoint of two given points (x1, y1) and (x2, y2), are given by:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
In this case, we want to find two points that divide the segment AB into three equal parts. Let's call these points P and Q.
First, we need to find the coordinates of the midpoint of AB.
Midpoint of AB(M) = ((-6 + 3) / 2, (-3 + 1) / 2)
= (-3/2, -1/2)
Now, to find the coordinates of P, we will use the midpoint formula with A as one of the endpoints and M as the other:
P = ((-6 + (-3/2)) / 2, (-3 + (-1/2)) / 2)
= ((-12 + (-3/2)) / 4, (-6 + (-1/2)) / 4)
= (-27/8, -13/8)
Similarly, to find the coordinates of Q, we will use the midpoint formula with M as one of the endpoints and B as the other:
Q = (((-3/2) + 3) / 2, ((-1/2) + 1) / 2)
= (((-3/2) + 6) / 4, ((-1/2) + 2) / 4)
= (15/8, 7/8)
Therefore, the coordinates of the points P and Q, which divide the segment AB into three equal parts, are P(-27/8, -13/8) and Q(15/8, 7/8).
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
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.
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.