Difference between isometry and euclidean motion I know that they both preserve distance however what is the difference between them?
I think isometry is part of Linear Transformations and euclidean motion is not
 A: Let me restrict attention to the Euclidean plane $\mathbb E^2$ (similar considerations play out in higher dimensions, although enumerating the Euclidean motions in those cases is more laborious).
The Euclidean motions of $\mathbb E^2$, by definition, consist of: the translations, rotations, reflections, and glide reflections, and the identity map. In the study of Euclidean geometry one often starts with just the reflections, but then one proves that the motions that you get by composing finitely many reflections consist precisely of the translations, rotations, reflections, glide reflections, and the identity. In more detail: the composition of an even number of reflections is always a translation, rotation, or the identity map; and the composition of an odd number of reflections is always a reflection or glide reflection.
The isometries of the plane, by definition, are all functions $f : \mathbb E^2 \to \mathbb E^2$ that preserve distance, meaning that $d(f(p),f(q))=d(p,q)$ for all $p,q \in \mathbb E^2$.
Now there's a theorem to prove:

In $\mathbb E^2$, every Euclidean motion is an isometry and every isometry is a Euclidean motion.

The hardest part of the proof is the following key step, which is kind of the last step in proving that every isometry is a Euclidean motion:

If an isometry fixes the three vertices of some triangle then that isometry is the identity map.

