Point Y lies on line segment XZ, between X and Z. The (x,y) coordinates for X and Z are (7,2) and (-3,-3), respectively. If the ratio of line segment XY to line segment YZ is 2 to 3, what are the (x,y) coordinates of Y?
To find the coordinates of point Y, we first need to determine the coordinates of the midpoint between X and Z.
The midpoint formula between two points (x1, y1) and (x2, y2) is given by:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Applying the formula using the coordinates of X (7,2) and Z (-3,-3):
Midpoint = ((7 + -3) / 2, (2 + -3) / 2)
Midpoint = (4/2, -1/2)
Midpoint = (2, -0.5)
Since the ratio of XY to YZ is 2 to 3, the coordinates of Y can be found by taking two-thirds of the distance between the midpoint and Z and adding it to the midpoint coordinates.
Let's calculate the x-coordinate of Y:
x-coordinate of Y = midpoint x-coordinate + (2/3) * (Z x-coordinate - midpoint x-coordinate)
x-coordinate of Y = 2 + (2/3) * (-3 - 2)
x-coordinate of Y = 2 + (2/3) * (-5)
x-coordinate of Y = 2 - (10/3)
x-coordinate of Y = 2 - 3.33
x-coordinate of Y ≈ -1.33
Now, let's calculate the y-coordinate of Y:
y-coordinate of Y = midpoint y-coordinate + (2/3) * (Z y-coordinate - midpoint y-coordinate)
y-coordinate of Y = -0.5 + (2/3) * (-3 - (-0.5))
y-coordinate of Y = -0.5 + (2/3) * (-2.5)
y-coordinate of Y = -0.5 + (-1.67/3)
y-coordinate of Y = -0.5 - 0.56
y-coordinate of Y ≈ -1.06
Therefore, the (x,y) coordinates of Y are approximately (-1.33, -1.06).
To find the coordinates of point Y, we need to determine its position on the line segment XZ. This can be done by dividing the segment XZ into two parts based on the given ratio of 2 to 3.
First, let's calculate the differences in the x and y coordinates between points X and Z:
Δx = Xz - Xx = -3 - 7 = -10
Δy = Yz - Yx = -3 - 2 = -5
Next, we need to determine the coordinates of point Y based on the ratio of 2 to 3:
Let's assume the coordinates of point Y as (x,y). If we consider point Y to be a certain fraction (2/5) of the total segment XZ and point Z to be the remaining fraction (3/5), we can write the following equations:
(x - 7) / (-10) = 2/5
(y - 2) / (-5) = 2/5
To solve these equations, we can cross-multiply:
5(x - 7) = -10 * 2
5(y - 2) = -5 * 2
Simplifying:
5x - 35 = -20
5y - 10 = -10
Adding 35 to both sides of the first equation and adding 10 to both sides of the second equation:
5x = 15
5y = 0
Dividing both sides of the equations by 5:
x = 15/5
y = 0/5
Therefore, the coordinates of point Y are (3, 0).
Y is 2/5 of the way from X to Z
x changes by -10
y changes by -5
when moving from X to Z. So,
Y = X+(-4,-2) = (3,0)