Question

Find the coordinates of point P along the directed line segment AB with endpoints A(3,2) and B(6,8) so that the ratio of AP to PB is 2:1.

Answer 1

The coordinates of point P, which divides the directed line segment AB with endpoints A(3,2) and B(6,8) in the ratio 2:1, are (5, 6).

Step-by-step explanation:

To find the coordinates of point P along the directed line segment AB with endpoints A(3,2) and B(6,8) such that the ratio of AP to PB is 2:1, we will use the concept of partitioning a line segment in a given ratio. Since the ratio of AP to PB is 2:1, we can use the section formula which is given by:

P(x, y) = ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n))

Where A(x1, y1), B(x2, y2), and m:n is the given ratio. Plugging in our values:

• A (3, 2) = (x1, y1)
• B (6, 8) = (x2, y2)
• m:n = 2:1

Therefore, we have:

P(x, y) = ((2*6 + 1*3)/(2 + 1), (2*8 + 1*2)/(2 + 1))

Simplifying this, we get:

P(x, y) = ((12 + 3)/3, (16 + 2)/3)

P(x, y) = (15/3, 18/3)

P(x, y) = (5, 6)

The coordinates of point P that divides the line segment AB in the ratio 2:1 are (5, 6).