Answer:
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:
[tex]|AB| =\sqrt{(53-(-7))^2+(-25-5)^2} = \sqrt{3600+900} =\sqrt{4500} =\sqrt{2^2\,3^2\,5^3} =5\,\sqrt{180}[/tex]
[tex]|AC| =\sqrt{(5-(-7))^2+(-1-5)^2} = \sqrt{144+36} =\sqrt{180}[/tex]
Therefore this new point B is indeed giving as a segment of the desired length.
