so suppose Fx = Mx + v, i.e. a potential scaling/rotation and a translation. It should be pretty clear that the translation has no effect on the length. So you just have to show what properties M must have to preserve lengths.
In the other direction suppose you have an isometry, then d(y1,y2)=d(x1,x2) and you can work out from there what linear transformations satisfy this and thus derive M and v.
The rest of it is just chasing the transformation through the process of deriving arc length, curvature, torsion.