Point A is at (-7,5) and point C is at (5,-1). Find the coordinates of point B on AC such that AB is 5 times as long as BC.

Point B = (53, -25)  satisfies the required condition

Step-by-step explanation:

Notice that point A is in the second quadrant and point C in the fourth quadrant, and the separation between them in the x-direction is given by:

| 5 - (-7) | = 12  

while their separation in the y-direction is:  

| -1 -5 | = 6

That means that point C is twelve units to the right and 6 units down from point A. In order to create a segment with another point aligned with them and which has 5 times the length of AC, we then need to consider adding to the x-coordinate of point C 4 times 12 units = 48 units to obtain the x-coordinate of the new point B, and go down 4 times 6 units = 24 units to the y-coordinate of point C.

This is: B = (5+48, -1-24) = (53, -25)

We can verify our procedure by estimating the length of the segment AB and that of segment AC and checking that the first one is indeed 5 times that of AC:

Therefore this new point B is indeed giving as a segment of the desired length.