If the point (x, y) is one-third of the way from (-2, 6) to (6,-8), then what is x + y?

What is the x+y?

for the x:

(x+2)/(6+2) = 1/3
3x + 6 = 8
3x = 2
x = 2/3

for the y:
(y-6)/(-8-6) = 1/3
3y - 18 = -14
3y = 4
y = 4/3

(x,y) = (2/3,4/3)

check: distance form (-2,6) to (2/3,4/3)
= √( (-2-2/3)^2 + (6-4/3)^2
= √( 64/9 + 196/3)
=√(260/9)
= (1/3)√260

distance from (-2,6) to 6,-8)
= √( 64 + 196)
= √260

so the distance checks out

x+y = 2/3 + 4/3 = 6/3 = 2

Well, let's do some math and clown around a bit!

First, let's find the halfway point between (-2, 6) and (6, -8).

The x-coordinate halfway between -2 and 6 is (-2 + 6)/2 = 4/2 = 2.
The y-coordinate halfway between 6 and -8 is (6 + -8)/2 = -2/2 = -1.

So, the halfway point is (2, -1).

Now, we need to find a point that is one-third of the way from (-2, 6) to (6, -8).

To do that, we can subtract one-third of the x distance and one-third of the y distance from the halfway point.

For the x-coordinate: 2 - (1/3)(6 - -2) = 2 - (1/3)(8) = 2 - (8/3) = (6/3) - (8/3) = -2/3.

For the y-coordinate: -1 - (1/3)(-8 - 6) = -1 - (1/3)(-14) = -1 - (-14/3) = (3/3) - (-14/3) = 17/3.

So, the point one-third of the way from (-2, 6) to (6, -8) is (-2/3, 17/3).

Finally, let's find the sum of the x and y coordinates: (-2/3) + (17/3) = (15/3) = 5.

So, x + y is equal to 5.

I hope this answer brought a smile to your face!

To find the point that is one-third of the way from (-2, 6) to (6, -8), we can use the concept of midpoint formula.

The midpoint formula is given by:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

In this case, (-2, 6) represents (x1, y1) and (6, -8) represents (x2, y2).

Let's calculate the midpoint:

Midpoint = (((-2) + 6)/2, (6 + (-8))/2)
= (4/2, -2/2)
= (2, -1)

Now, to find the point that is one-third of the way, we need to divide the difference between the two points by 3 and add it to the first point.

Difference between x-coordinates = 6 - (-2) = 8
Difference between y-coordinates = -8 - 6 = -14

So, dividing the difference by 3 gives us:

x-coordinate difference = 8/3
y-coordinate difference = -14/3

Adding the x-coordinate difference to the x-coordinate of the first point:

x = -2 + (8/3) = (-6/3) + (8/3) = 2/3

Adding the y-coordinate difference to the y-coordinate of the first point:

y = 6 + (-14/3) = (18/3) + (-14/3) = 4/3

Therefore, the point (x, y) that is one-third of the way from (-2, 6) to (6, -8) is (2/3, 4/3).

To find x + y:

x + y = (2/3) + (4/3) = 6/3 = 2

To find x and y, we can use the concept of finding the midpoint between two points. The midpoint formula is given by:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

In this case, we have the two given points (-2, 6) and (6, -8). To find the point that is one-third of the way from (-2, 6) to (6, -8), we need to find the midpoint of these two points and then find the coordinates one-third of the way between them.

Let's calculate the midpoint first:

Midpoint = ((-2 + 6) / 2, (6 + (-8)) / 2)
= (4 / 2, -2 / 2)
= (2, -1)

Now, let's find the coordinates one-third of the way from (-2, 6) to (6, -8) using the given midpoint. We can calculate it as follows:

One-third point = ((2 + (-2)) / 3, (-1 + 6) / 3)
= (0 / 3, 5 / 3)
= (0, 5/3)

Finally, we need to find x + y for the point (0, 5/3):

x + y = 0 + 5/3
= 5/3

So, x + y equals 5/3.