I have a problem that asks me to
Find all length preserving transformations of the plane that send point A to point A’ and point B to point B’ where: $A=(0,1), B=(1,1), A’=(3,2), B’=(3- \frac{\sqrt3}{2}, \frac{3}{2})$;
and to write the transformations as a parallel transport followed by a rotation about the origin, and possibly a reflection.
I did some preliminary work and found that for the rotation, $\theta = \frac{\pi}{6}$. I’m now left with systems of equations that involve the variables of transport. Would it just remain to solve the system for those variables? And how can I determine if a reflection is needed? Do I need to take into account the possible reflection when I write the formula for the points after translation and rotation?
Consider in a separate way rotations and symmetries (symmetries with respect to a line of course ; symmetries with respect to a point are merely $\pi$-rotations) ; but you haven't to mix them, for example, you haven't to consider a rotation followed by a symmetry (because it is a symmetry...).
$$\binom{x'}{y'}=\pmatrix{\cos a&-\sin a\\ \sin a& \ \ \cos a}\left(\binom{x}{y}+\binom{u}{v}\right)$$
in which you will "plug" the conditions $A(x,y) \to A'(x',y')$, $B(x,y) \to B'(x',y')$ giving 4 (compatible!) equations with 3 unknowns $a,u,v$.
$$\binom{x'}{y'}=\pmatrix{\cos (2b)&\ \ \ \ \ \sin (2b)\\ \sin (2b)& \ \ -\cos (2b)}\left(\binom{x}{y}+\binom{u}{v}\right)$$
(see here for explanations).
I wouldn't be surprized indeed that angles a and $b$ are positive or negative multiples of $\pi/6$ or $\pi/12$.