Find the coordinates of the point that lies one fourth of the way from (2,-10) to (6,0)?

Question

Im having a hard time figuring this one out can someone help

Answer 1

use the midpoint formula twice
let (x1,y1)=(2,-10) and (x2,y2)=(6,0)
the midpoint is
( (x1+x2)/2 , (y1+y2)/2 )
( (2+6)/2 , (-10+0)/2 )
=(4,-5) then call this the new (x2,y2)
gives
( (2+4)/2 , ( -10+ -5)/2 )
=(3, -15/2)

alternatively you could use the weighted average
think of ( (x1+x2)/2 , (y1+y2)/2 )
as (1/2*x1 +1/2*x2 , 1/2*y1 + 1/2*y2 )
the x1 and x2 are weighted evenly ie 1/2 each
you want to weight x1 more strongly than x2
ie weight x1 with 3/4 and x2 with 1/4
and similarly for y1 and y2
this gives the formula
(3/4*x1 +1/4*x2 , 3/4*y1 + 1/4*y2 ) gives
(3/4*2 +1/4*6 , 3/4*-10 + 1/4*0 )
=(3, -15/2) as before
this way is better since it can be applied
to any fraction, just note that the weights must add to 1

the end
.

Answer 2

(3, -15/2) are the coordinates of the point you are looking for.

To find the answer to this, just add 1/4 of the linear change of the x coordinates and the y coordinates to the smaller of the x and y values, x = 2 and y = -10. We add the change to the smaller values because we are moving from one fixed point to a new point which lies in the positive direction from the first. Hence, the change will be in the positive direction, which implies addition.

For the x values: (6 - 2)/4 = 1
x' = 2 + 1 = 3

For the y values: [0 - (-10)] / 4 = 10/4 = 5/2
y' = -10 + 5/2
y' = (-20 + 5)/2
y' = -15/2

If we were moving from (6, 0) to (2, -10) we would be moving in the negative direction, hence the change would be negative, which implies we would subtract from x = 6 and y = 0.

x' = 6 - 1 = 5
y' = 0 - (5/2) = -5/2

Answer 3

If you haven't done it, then use some grid paper and try to figure it out again. After a while you won't have to use grid paper because you will see the coordinate plane in your head when you are given one of these questions.

Here's was my approach:

The problem says "from (2,-10) to (6,0)" so we know we need to move one forth of the total distance away from (2, -10). But what is the total distance?

Well, to think about the total distance, we remember that we are beginning at (2,-10) and then ending at (6,0). So, if we start on 2 and end at 6, that distance is 4. And if we begin at -10 and end at 0 that distance is 10.

Look at our total distances and find one forth of them.
One forth of 4 is 1. One forth of 10 is 2.5.

Now begin at (2, -10) and think about the movement, 1 in the positive direction away from 2 is 3 and 2.5 in the positive direction away from -10 is -7.5. So, we are left at (3, -7.5).

Grid paper!