Question

Find the segment AB so that AP to PB is the given ratio. Find the coordinates of point P along the directed line.

A(1, 3), B(8, 4); 4 to 1

Answer 1

To find the segment AB so that the ratio of AP to PB is given, divide the distance from A to B in the given ratio. The segment AB so that the ratio of AP to PB is 4 to 1 is (5/4 * sqrt(2)) units long.

Step-by-step explanation:

To find the segment AB so that the ratio of AP to PB is given, we need to divide the distance from A to B in the given ratio. In this case, the ratio is 4 to 1. So, we can divide the distance from A to B into 4 equal parts and take 1 of those parts.

First, let's find the distance from A to B using the distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2). In this case, A(1, 3) and B(8, 4). Plugging these values into the formula, we get: sqrt((8 - 1)^2 + (4 - 3)^2) = sqrt(49 + 1) = sqrt(50) = 5sqrt(2)

Now, divide the distance 5sqrt(2) into 4 equal parts: (5sqrt(2))/4 = 5/4 * sqrt(2)

So, the segment AB so that the ratio of AP to PB is 4 to 1 is (5/4 * sqrt(2)) units long.