Find the coordinates of the point which is 3/5 of the way from (3, 2) to (-3, 5).
(3 + (2/3)(-3-3),(2 + (2/3)(5-2))
= (3-4,2+2)
= (-1,4)
To find the coordinates of the point which is 3/5 of the way from (3, 2) to (-3, 5), we need to consider the x and y coordinates separately.
Step 1: Calculate the change in x and y coordinates:
Δx = (x2 - x1) = (-3 - 3) = -6
Δy = (y2 - y1) = (5 - 2) = 3
Step 2: Multiply Δx and Δy by 3/5:
Δx * (3/5) = -6 * (3/5) = -18/5
Δy * (3/5) = 3 * (3/5) = 9/5
Step 3: Add the result from step 2 to the initial coordinates (3, 2):
x = x1 + (Δx * (3/5)) = 3 + (-18/5) = -3/5
y = y1 + (Δy * (3/5)) = 2 + (9/5) = 19/5
Therefore, the coordinates of the point which is 3/5 of the way from (3, 2) to (-3, 5) are (-3/5, 19/5).
To find the coordinates of a point that is a certain fraction of the way from one point to another, we can use the concept of linear interpolation.
Let's denote the coordinates of the starting point as (x1, y1) and the coordinates of the ending point as (x2, y2). In this case, the starting point is (3, 2) and the ending point is (-3, 5).
The formula for linear interpolation is:
x = (1 - t) * x1 + t * x2
y = (1 - t) * y1 + t * y2
Where 't' represents the fraction (or proportion) of the distance between the two points.
In this case, you mentioned the point is 3/5 (or 0.6) of the way.
Substituting the given values into the formula, we have:
x = (1 - 0.6) * 3 + 0.6 * (-3)
y = (1 - 0.6) * 2 + 0.6 * 5
Simplifying the equations:
x = 0.4 * 3 - 0.6 * 3
y = 0.4 * 2 + 0.6 * 5
x = 1.2 - 1.8
y = 0.8 + 3
x = -0.6
y = 3.8
Therefore, the coordinates of the point that is 3/5 of the way from (3, 2) to (-3, 5) are (-0.6, 3.8).