Question

The directed line segment from point A(-9, 2) to point B(12, 8) is divided by the point P(-2, 4) in the ratio m:n. What ratio represents AP:PB?

a) 5:3
b) 3:5
c) 2:7
d) 7:2

Answer 1

The ratio representing AP:PB for the points A, B, and P given is 7:2. This is determined by applying the section formula for a line divided in a given ratio to find the ratio in which point P divides segment AB.

Step-by-step explanation:

The question provided is concerned with the division of a directed line segment from point A(-9, 2) to point B(12, 8) by point P(-2, 4) in a certain ratio. The ratio in question represents the lengths of segment AP to segment PB. To calculate this we will use the section formula in two dimensions, which states:

• For a given ratio m:n, the coordinates of point P dividing the line segment in the ratio m:n are given by ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)), where x1, y1 and x2, y2 are coordinates of points A and B, respectively.

Applying the formula, we get the following equations for the coordinates of P:

1. -2 = ((m * 12) + (n * -9)) / (m + n)
2. 4 = ((m * 8) + (n * 2)) / (m + n)

On simplifying the equations, we find that m:n equals 7:2, which means the ratio of AP to PB is 7:2, corresponding to option (d).