Show that a map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is an isometry for the Euclidean metric on the plane $\mathbb{R}^{2}$ if and only if there is a vector $\boldsymbol{v} \in \mathbb{R}^{2}$ and an orthogonal linear map $B \in \mathrm{O}(2)$ with

$$T(\boldsymbol{x})=B(\boldsymbol{x})+\boldsymbol{v} \quad \text { for all } \boldsymbol{x} \in \mathbb{R}^{2}$$

When $T$ is an isometry with $\operatorname{det} B=-1$, show that $T$ is either a reflection or a glide reflection.