Geometry Transformations

Question

How do you prove that transformation $(x,y)=(-x,y+2)$ is an isometry?

Not sure where to start. I know this means $x'=-x$ and $y'=y+2$ but what do you use for the points to calculate the image points?

Answer 1

Isometry means it keeps distances, i.e. for all $x_1,x_2,y_1,y_2 \in \mathbb R$, $$\|(x_1,y_1) - (x_2,y_2)\| = \| (-x_1,y_1+2) - (-x_2,y_2+2) \|$$ Is that true in this case?

You can also write this transformation as the composition of a reflection and a translation, and use that both of those are isometries, therefore so is their composition.

Answer 2

The title just says geoemtry, so here is an approach from differential geometry, which is definitely overkill (maybe). If this is an isometry then the first fundamental form $\textbf{I}(\sigma)$ must agree with that on the plane i.e $du^2+dv^2$. Here $\sigma(u,v) = (-u,v+2)$ and so $\|\sigma_u\|^2 = 1= \|\sigma_v\|^2$ and we have $\sigma_u \cdot \sigma_v = 0$. Hence, $\textbf{I}(\sigma) = du^2 + dv^2$ i.e $\sigma$ is an isometry.