Final answer:
The coordinates of point P, which is (3/4) of the way along the directed line segment from (6, -5) to (-3, 4), are calculated using the section formula and the ratio 3:1. By applying the values to the formula, the coordinates of point P are found to be (-3/4, 7/4).
Step-by-step explanation:
To find the coordinates of point P that is (3/4) of the way along the directed line segment from (6, -5) to (-3, 4), you can use the formula for finding a point that divides a line segment in a given ratio. In this case, the ratio is 3:1. The formula for a point P(x, y) dividing line segment AB with A(x1, y1) and B(x2, y2) in the ratio m:n is:
To find the coordinates of point P that is (3/4) of the way along the directed line segment from (6, -5) to (-3, 4), we can use the concept of midpoint formula.
The coordinates of the midpoint M between two points (x_1, y_1) and (x_2, y_2) can be found using the formulas:
x = (x_1 + x_2)/2 and y = (y_1 + y_2)/2
In this case, the coordinates of the midpoint M are x = (6 + -3)/2 = 3/2 and y = (-5 + 4)/2 = -1/2.
To find the coordinates of point P, which is 3/4 of the way from M to (-3, 4), we can use the formula:
x = x_1 + (3/4)(x_2 - x_1) and y = y_1 + (3/4)(y_2 - y_1)
Plugging in the values x_1 = 3/2, x_2 = -3, y_1 = -1/2, and y_2 = 4, we can calculate:
x = (3/2) + (3/4)(-3 - 3/2) = -3/8 and y = (-1/2) + (3/4)(4 - (-1/2)) = 7/8.
Therefore, the coordinates of point P are (-3/8, 7/8).
P = ( (mx2 + nx1) / (m + n), (my2 + ny1) / (m + n) )
By substituting the given points A(6, -5) and B(-3, 4), and the ratio m:n = 3:1, we get:
P = ( (3(-3) + 1(6)) / (3 + 1), (3(4) + 1(-5)) / (3 + 1) )
P = ( (-9 + 6) / 4, (12 - 5) / 4 )
P = ( -3/4, 7/4 )
So, the coordinates of point P are (-3/4, 7/4).