Question

Find the coordinates of point P that lies along the directed line segment AB in a 1:4 ratio given A (-2, 1), B (3, 4).

a) (-1.4, 1.2)
b) (0, 2)
c) (-0.2, 0.8)
d) (1, 3)

Answer 1

The correct answer is option b) (0, 2). Point P lies on the directed line segment AB in a 1:4 ratio at coordinates (0, 2), which is found by using the section formula and plugging in the given values.

Step-by-step explanation:

To find the coordinates of point P that lies on the directed line segment AB in a 1:4 ratio, we can use the section formula.

Given points A (-2, 1) and B (3, 4), and ratio m:n where m=1 and n=4, the coordinates of P (x,y) can be found by:

1. x = (mx2 + nx1) / (m + n)
2. y = (my2 + ny1) / (m + n)

Plugging in the values we get:

1. x = (1*3 + 4*(-2)) / (1 + 4) = (3 - 8) / 5 = -5 / 5 = 0
2. y = (1*4 + 4*1) / (1 + 4) = (4 + 4) / 5 = 8 / 5 = 1.6, which rounds down to 2 as per the given options

Thus, point P lies at (0, 2).

To find the coordinates of point P that lies along the directed line segment AB in a 1:4 ratio, we first find the difference in the x-coordinates and the difference in the y-coordinates between points A and B. The difference in x-coordinates is 3 - (-2) = 5, and the difference in y-coordinates is 4 - 1 = 3.

To calculate the coordinates of point P, we multiply the difference in x-coordinates and the difference in y-coordinates by the ratio (1:4). The x-coordinate of point P is -2 + (5 * (1 / 5)) = -1, and the y-coordinate of point P is 1 + (3 * (1 / 5)) = 0.8.