Concept guide
Geometric transformations: proving the basic isometries and classifying uniform scaling
On the equal-unit orthonormal real plane, prove that translations, the counterclockwise quarter-turn R90(x,y)=(−y,x) about the origin, and the other rotations and axis reflections listed in the MKO are isometries; that the half-turn (−x,−y) is not a reflection in any of the listed axes; and that uniform scaling Hc(x,y)=(cx,cy) multiplies every distance by |c|. Use this to classify c=0, c=−1, c=−2 and c=2, with a numerical check on P=(3,1), Q=(−1,4).