1. The problem statement, all variables and given/known data Verify that the following mappings are isometries on R^2 Reflection Through the Origin Translation Rotation 2. Relevant equations Qualities of a metric: d(x,y) = d(y,x) d(x,x) = 0 d(x,y) = 0 <=> x = y d(x,y) =< d(x,z) +d(z,y) 3. The attempt at a solution As a metric hasn't been specified, I have been trying to prove this for a general metric using just the intrinsic qualities. I haven't had much luck, though. I know that all three are straightforward to prove in Euclidean Space, which gives a metric. But is there a simple proof for a general metric? I may have misunderstood the meaning of Verify, but would nevertheless like a proof if there is one.