To find the rule for the translation that maps P'Q' back to PQ, we need to consider the inverse transformation.
Given the original translation:
(x, y) → (x + m*y + n)
To find the inverse transformation, we want to go from P'Q' back to PQ. So, we need a rule that undoes the changes made to the coordinates of P' and Q'.
For point P' (x', y') and Q' (x'', y''), we can use the inverse transformation rule:
(x', y') → (x' - m*y' - n)
(x'', y'') → (x'' - m*y'' - n)
Applying this rule to P'Q' would effectively reverse the translation and bring us back to the original points PQ.
The reasoning behind this is that if the original transformation added m*y and n to the x-coordinate and y-coordinate, respectively, then the inverse transformation should subtract m*y and n from the x-coordinate and y-coordinate to reverse the changes and return to the original points.