Given XZ with X(-4,3) and Z (-6,-2) find the coordinates of Y if Y divides XZ one-fifth of the way from X to Z
To find the coordinates of point Y, which divides the line segment XZ one-fifth of the way from X to Z, we can use the concept of midpoint formula.
The midpoint formula states that the coordinates of the midpoint (M) between two points (X and Z) can be calculated by taking the average of their x-coordinates and the average of their y-coordinates.
Let's find the midpoint (M) of line segment XZ:
x-coordinate of midpoint (M) = (x-coordinate of X + x-coordinate of Z) / 2
= (-4 + -6) / 2
= -10 / 2
= -5
y-coordinate of midpoint (M) = (y-coordinate of X + y-coordinate of Z) / 2
= (3 + -2) / 2
= 1 / 2
= 0.5
So the midpoint (M) of line segment XZ is (-5, 0.5).
To find the coordinates of Y, which is one-fifth of the way from X to Z, we can use the following formulas:
x-coordinate of Y = (4/5) * x-coordinate of X + (1/5) * x-coordinate of Z
= (4/5) * -4 + (1/5) * -6
= -3.2 - 1.2
= -4.4
y-coordinate of Y = (4/5) * y-coordinate of X + (1/5) * y-coordinate of Z
= (4/5) * 3 + (1/5) * -2
= 2.4 - 0.4
= 2
So the coordinates of point Y are (-4.4, 2).
To find the coordinates of point Y, which divides the line segment XZ into the ratio 1:5, we can use the concept of section formula.
The section formula states that if a line segment with endpoints A(x1, y1) and B(x2, y2) is divided by a point P (x, y) such that the ratio of AP to PB is m:n, then the coordinates of P can be found using the following formula:
x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)
In this case, the ratio of AP to PB is 1:5, which means m = 1 and n = 5.
The coordinates of X are (-4, 3) and the coordinates of Z are (-6, -2).
Using the section formula, we can calculate the coordinates of Y:
x = (1*(-6) + 5*(-4)) / (1 + 5)
= (-6 - 20) / 6
= -26 / 6
= -13 / 3
y = (1*(-2) + 5*3) / (1 + 5)
= (-2 + 15) / 6
= 13 / 6
Therefore, the coordinates of point Y are (-13/3, 13/6).
-6-(-4) = -2
-2-3 = -5
So, adding 1/5 of those distances to X in both dimensions,
Y = (-4 + 1/5 * -2, 3 + 1/5 * -5) = (-4 2/5 , 2)
Actually, I suspect a typo, but you can be the judge of that.