Question

Find point Z that partitions the directed line segment XY in the ratio of 5:3 where X(-2, 6) and Y(-10, -2).

Its by using the part to part method

Answer 1

I have no idea what "the part to part method" method is, but I taught it like this:

I will assume that YZ : ZX = 5:3

for the x of Z:
(x - (-10)) / (-2 - x) = 5/3
(x+10)/(-2-x) = 5/3
3x + 30 = -10 - 5x
8x = -40
x = -5

for the y of Z:
(y+2)/(6-y) = 5/3
3y + 6 = 30 - 5y
8y = 24
y = 3

so Z = (-5,3)

Answer 2

divide XY in the ratio of 5:3 where X(-2, 6) and Y(-10, -2).
You want Z to be 5/8 of the way from X to Y.
The distance from -2 to -10 is -8; 5/8 of that is -5
The distance from 6 to -2 is also -8, so 5/8 of that is -5
Z = (-2,6)+(-5,-5) = (-7,1)