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
To find the coordinates of point Z that partitions the directed line segment XY in the ratio of 5:3, we can use the part-to-part method.
First, we need to calculate the total change in x and y coordinates between points X and Y:
Δx = X-coordinate of Y - X-coordinate of X = (-10) - (-2) = -10 + 2 = -8
Δy = Y-coordinate of Y - Y-coordinate of X = (-2) - 6 = -2 - 6 = -8
Next, we need to find the part of the total change that corresponds to the ratio 5:3:
To find the part corresponding to the ratio 5:3, we need to divide the total change by the sum of the ratio parts, which is 5 + 3 = 8.
Using this information, we can calculate the part of the total change for the x-coordinate and y-coordinate:
Part of Δx = (5/8) * Δx = (5/8) * (-8) = -5
Part of Δy = (5/8) * Δy = (5/8) * (-8) = -5
Finally, we need to add the part of the total change to the coordinates of point X to find the coordinates of point Z:
X-coordinate of Z = X-coordinate of X + Part of Δx = -2 + (-5) = -7
Y-coordinate of Z = Y-coordinate of X + Part of Δy = 6 + (-5) = 1
Therefore, the coordinates of point Z that partitions the directed line segment XY in the ratio of 5:3 are Z(-7, 1).
To find point Z that partitions the line segment XY in the ratio of 5:3, we can use the section formula. This formula calculates the coordinates of a point that divides a line segment in a given ratio.
The section formula is:
X = [(x2 × m) + (x1 × n)] / (m + n)
Y = [(y2 × m) + (y1 × n)] / (m + n)
In this case, we are given the following:
Point X: X(-2, 6)
Point Y: Y(-10, -2)
Ratio: 5:3
To apply the section formula, we need to assign weights to the ratio. Let's say the weight for 5 is 'm' and the weight for 3 is 'n'. Therefore, m = 5 and n = 3.
Now, let's substitute the given values into the formula:
X = [(-10 × 5) + (-2 × 3)] / (5 + 3) = (-50 - 6) / 8 = -56 / 8 = -7
Y = [(-2 × 5) + (6 × 3)] / (5 + 3) = (-10 + 18) / 8 = 8 / 8 = 1
Hence, the coordinates of point Z that partitions the line segment XY in the ratio of 5:3 are Z(-7, 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)
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)