Let $ρθ$ be the counterclockwise rotation about the origin $(0, 0)$ through the angle $θ$.
1) Give a proof in the 2-dimensional real plane that $ρθ$ is an isometry.
I am confused if I need to show that $d(p_\theta (x,y), (0,0)) = d((x,y), (0,0)) $ or if I need to pick two different arbitrary points and show that $d(p_\theta (x,y),p_\theta (r,s)) = d((x,y), (r,s)) $.
2) Prove that every general rotation $ρ$ of $R^2$ is an isometry of $R^2$
My thinking for this one is that I would be to show that given a fixed point $(r,s)$ and an arbitrary $(x,y)$, $d(p_\theta (x,y), (r,s)) = d((x,y), (r,s)) $. I'm not sure how to go about proving this.
Any help would be appreciated.
1) An isometry preserves the distance between every pair of points, so you need to show $d(\rho_\theta(P_1),\rho_\theta(P_2))=d(P_1,P_2)$ for an arbitrary pair of points.
2) Decompose the general rotation into a sequence of transformations that you already know to be isometries (you'll use the result from part 1).