given the point A(-5,-3) and B(4,9), find the coordinates of the point Pon dirrected line segment AB that partitions AB into a ratio of 1 to 2
so P is 1/3 of the way from A to B
p(-2,1)
To find the coordinates of the point P that partitions line segment AB into a ratio of 1 to 2, we need to calculate the coordinates of the point that is one-third of the way from point A to point B.
Step 1: Find the difference in x-coordinates and y-coordinates between points A and B.
Δx = x-coordinate of B - x-coordinate of A = 4 - (-5) = 9
Δy = y-coordinate of B - y-coordinate of A = 9 - (-3) = 12
Step 2: Divide the difference in x-coordinates and y-coordinates by 3.
Δx/3 = 9/3 = 3
Δy/3 = 12/3 = 4
Step 3: Add Δx/3 to the x-coordinate of point A and Δy/3 to the y-coordinate of point A.
x-coordinate of P = x-coordinate of A + Δx/3 = -5 + 3 = -2
y-coordinate of P = y-coordinate of A + Δy/3 = -3 + 4 = 1
Therefore, the coordinates of point P on the directed line segment AB that partitions AB into a ratio of 1 to 2 are P(-2, 1).
To find the coordinates of point P on the directed line segment AB, we need to determine the ratio in which AB is divided.
The ratio of 1 to 2 means that the segment AB is divided into three equal parts: one part for the first ratio, and two parts for the second ratio.
First, let's calculate the total change in the x-direction and y-direction between points A and B:
x-change: Bx - Ax = 4 - (-5) = 9
y-change: By - Ay = 9 - (-3) = 12
Next, we need to determine the change in the x-direction and y-direction for the first ratio. Since the first ratio is 1 and the total change is 3 equal parts, the change for the first ratio will be 1/3 of the total change:
x-change for first ratio: (1/3) * 9 = 3
y-change for first ratio: (1/3) * 12 = 4
Now, we can find the coordinates of point P by starting from point A and adding the changes in the x-direction and y-direction for the first ratio:
Px = Ax + x-change for first ratio = -5 + 3 = -2
Py = Ay + y-change for first ratio = -3 + 4 = 1
Therefore, the coordinates of point P on the directed line segment AB that partitions AB into a ratio of 1 to 2 are P(-2, 1).