Final answer:
Without specific figures, it is not possible to provide exact transformations or coordinate notations, but transformations include reflections, translations, rotations, and dilations. Distance is invariant under rotations due to the mathematical properties of the equations that describe such transformations.
Step-by-step explanation:
To determine the sequence of similarity transformations that map triangle AABC to triangle APQR without specific figures to reference, it is not possible to provide the coordinate notation for each transformation or the exact sequence. Usually, in high school geometry, the sequence of transformations to map one figure onto another could include any combination of reflections, translations, rotations, and dilations.
For example, a reflection might involve flipping the figure across a line, a translation would slide the figure from one place to another without rotating or resizing it, a rotation would turn the figure around a fixed point, and a dilation would resize the figure by scaling it larger or smaller while maintaining its shape.
The invariant property of distance under rotations can be shown if the rotation transformation equations are given as x' = x cos θ + y sin θ and y' = -x sin θ + y cos θ. This implies that the distance between any two points remains the same after the rotation because the transformation preserves distances.