Final answer:
The correct transformation that maps PQ to P'Q' is reflection across the x-axis. This is determined by the fact that only the y-coordinates have changed signs, whereas under rotations, both coordinates would be affected differently.
Step-by-step explanation:
To determine which transformation maps PQ to P'Q', we analyze the given points. P(2,2) transforms to P'(2,-2) and Q(6,4) to Q'(6,-4). A quick observation shows that the y-coordinates of both P and Q have been multiplied by -1 while their x-coordinates remain unchanged. This transformation is characteristic of a reflection across the x-axis.
For rotation, the coordinates change according to x' = x cos(θ) + y sin(θ) and y' = -x sin(θ) + y cos(θ). The point P undergoes a 90° or 270° counterclockwise rotation about the origin would result in coordinates that do not match P'. Similarly, a reflection across the y-axis would change the x-coordinates of the points, which is not the case here. Therefore, the correct transformation is option 4, reflection across the x-axis.
The invariance of distance after transformation can be demonstrated algebraically. For rotations, the distance of a point P(x,y) to the origin (0,0) is given by √(x2 + y2) and this value does not change with rotation because the new coordinates (x',y') also satisfy x'2 + y'2 = x2 + y2. Similarly, the distance between two points P(x1,y1) and Q(x2,y2), given by √((x2-x1)2 + (y2-y1)2), is also invariant under rotations.