Remark
If you can, draw the original diagram of the line segment AB. Make it about 1 inch long. Some distance away, make a line segment that is about 1.8 inches long and parallel to AB. That is going to be A'B'. Now when you draw A'B' It is going to have the same slope as AB which is - 1
Discussion
Here's the catch to this problem. B is the right angle. Is similarity retained when you dilate from the origin. The answer is yes. See the diagram below. The key to this question is understanding that similarity is retained. It it is, then the slopes are the same
The second thing you have to understand is that the right angle is at B. That mean that AB and BC meet at right angles. That means that the slope of BC is
m1 * m2 = -1
where m1 is the slope of AB and m2 is the slope of BC. But we are told that AB has a slope of -1
Therefore m1 = -1
-1 * m2 = -1
m2 = 1
so BC has a slope of 1. Now for the answer.
If BC has a slope of 1 slope is maintained by similarity, then B'C' has a slope of 1 which is your answer
Problem 2
The length you have given is the length of the Hypotenuse. you are asked for AE.
You need to set up a proportion for this question.
You know the most about the small triangle so start with it
AD / AB = (AD + DE) /(AB + BC)
Givens
AD = 5200
DE = x
AB = 5600
AC = 5600 + 7000 = 12600
Solve
5200/(5200 + x) = 5600 / 12600 Cross multiply
5200 * 12600 = (5200 + x) * 5600 Multiply the left and remove the brackets on the right.
65520000 = 29120000 + 5600x Subtract 29120000 from both sides.
65520000 - 29120000 = 5600x
35400000 = 5600x Divide by 5600 on both sides.
36400000 / 5600 = x
6500 = x
Answer
AE = 5200 + 6500
AE = 11700 <<<< Answer
This should be smaller than 12600 which it is.
