Find the two motions $\Bbb E ^2 \to \Bbb E^2 $ taking $(0,0)\to (1,2)$ and $(0,\sqrt{2}) \to (2,3)$. Write each as $x \to Ax+b$ .Express them as a rotation and glide.
So far I only have that if $(0,0) \to A(0,0)+b=(1,2)$ , $b= (1,2)$.
But then $(0,\sqrt{2}) \to A(0,\sqrt{2}) +(1,2)=(2,3)$.
So $A(0,\sqrt{2}) =(1,1)$. But you can't find matrix $A$ with this since we have a $0$ in $(0,\sqrt{2}) $. I don't really know what to do
Hint. You can take $A$ as the rotation matrix: $$A:=\left( \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right)$$ Are you able to find $\theta$?
P.S. Another way (see Lee Mosher's comment below) is to take $A$ as the composition of a rotation and a glide reflection $$A= \left( \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right)\left( \begin{array}{cc} 1 &0 \\ 0 & -1 \end{array}\right).$$