How can a translation preserve a norm?

Question

I want to demonstrate that arc length, curvature and torsion of a parametrized curved are invariant under a rigid motion. However, when trying to do so for the arc length I get $$\left\|\frac{d\alpha}{dt}\right\|=\left\|\frac{d(A\circ\rho \alpha)}{dt}\right\|=\left\|(A\circ\rho) \frac{d\alpha}{dt}\right\| $$ Where $\alpha$ is the parametrized curve, A is a translation, defined as $A(\alpha)=\alpha+p$, with $p\in R^3$ and $\rho$ is an orthonormal transformation with positive determinant. I get that $\rho$ preserves norm, but how can a translation also preserve it? Supposedly, the norm of the vector after the transformation is $$\|A(\alpha)\|=\sqrt{(\alpha+p)\cdot(\alpha+p)}\neq\|\alpha\|$$ What am I missing?