Answer: To find the coordinates of the point that partitions the directed line segment from (-4, 2) to (-1, -4) into a ratio of 1:5, we can use the following steps:
Step 1: Find the difference between the x-coordinates and y-coordinates of the endpoints.
Difference in x-coordinates = -1 - (-4) = 3
Difference in y-coordinates = -4 - 2 = -6
Step 2: Determine the coordinates of the point that partitions the segment into a ratio of 1:5.
Let (x, y) be the coordinates of the point. Since the segment is being partitioned into a ratio of 1:5, the distance from (-4, 2) to (x, y) is one-sixth of the distance from (-4, 2) to (-1, -4). Therefore, we have:
Distance from (-4, 2) to (x, y) = (1/6) * Distance from (-4, 2) to (-1, -4)
Using the distance formula, we can find the distance between the two points:
Distance from (-4, 2) to (-1, -4) = sqrt(((-1) - (-4))^2 + ((-4) - 2)^2) = sqrt(3^2 + (-6)^2) = sqrt(45)
So, the distance from (-4, 2) to (x, y) is:
Distance from (-4, 2) to (x, y) = (1/6) * sqrt(45) = sqrt(5/2)
To find the coordinates of (x, y), we need to move a distance of sqrt(5/2) in the x-direction and a distance of 5 * sqrt(5/2) in the y-direction from (-4, 2). Since the difference in x-coordinates is 3, we can write:
x = -4 + 3 * (sqrt(5/2)/sqrt(45)) = -4 + sqrt(5/2)
Similarly, since the difference in y-coordinates is -6, we can write:
y = 2 + (-6) * (5 * sqrt(5/2)/sqrt(45)) = -4 - 5 * sqrt(5/2)
Therefore, the coordinates of the point that partitions the directed line segment from (-4, 2) to (-1, -4) into a ratio of 1:5 are:
(x, y) = (-4 + sqrt(5/2), -4 - 5 * sqrt(5/2))
Explanation: