If a line ⎯⎯⎯⎯⎯⎯⎯⎯ is translated in a plane to form →′′, what is true about → and →′′? AA’ = BB’

AA = BB

AB = BB

A’A’ = B’B’

AA’ = BB’

Step-by-step explanation:

Suppose that we have a point (x, y), and we apply a transformation that translates this point in a given direction, let's suppose that we translate n units in the horizontal axis, and m units in the vertical axis.

Then the new coordinates of the point will be (x+ n, x + m)

Notice that for any point (x, y) the distance between the original point and the translated point (x + n, x + m) is always the same.

D = √( (x - (x + n))^2 + (y - (y + m))^2)

D = √ ( n^2 + m^2)

Here you can see that there is no dependence with the original coordinates of the point.

So if we have a line and we translate it with a translation like the one defined above, the distance between all the points in the original line and the corresponding point in the translated line is always the same.

Then if we have the point A, and after the translation, it becomes A', the length of the segment AA' is equal as the distance between A and A'.

And exactly the same happens for B and B'.

Then if the distance between A and A', is the same as the distance between B and B', then:

AA' = BB'

2(2x + 3) = -4 + 2(2x + 5)

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 6

4x - 6 + 6 = 4x - 6 + 6

4x = 4x

x = x

unlimited solutions.