Relationship between definition of the Euclidean metric and the proofs of the Pythagorean theorem When working in $R^2$, we usually define the Euclidean length of a vector $(x, y)$ to be $\sqrt{x^2 + y^2}$. This is obviously related to the Pythagorean theorem -- if we view $(x, y)$ as the hypotenuse of a right triangle with legs $(x, 0)$ and $(0, y)$, the Euclidean length we just defined for this vector is of course is equal to the length of the hypotenuse of the corresponding right triangle, given by Pythagoras.
But why are we defining length in $R^2$ with this formula at all, isn't it already given by the theorem? 
I understand that we might want to use a different, non-Euclidean metric as our notion of distance in many contexts, and that the Pythagorean theorem doesn't generalize to equally valid, non-euclidean geometries. But then, from where does the proof of the Pythagorean theorem arise? That is, how can we "prove" the length of a right triangle is the sum of the squares of its legs if we haven't already defined distance in $R^2$ in the same way? What do we mean by "length" in the proof of the theorem, if we haven't already defined it accordingly?
I'm not asking to see a proof of the Pythagorean theorem. Rather, I'm asking what these proofs rest on -- are they less "formal" proofs than they are sources of geometric intuition for the Euclidean metric? Or is there some way we can prove the Pythagorean theorem before we have such a metric at all? Or am I asking the wrong questions?
Sorry if this has been asked before, searching turned up nothing helpful. I hope it's clear what I'm asking, please let me know if not.
 A: As you have mentioned in your question, we have different metrics in $\mathbb{R^2}$  as far as  metric topology  is concerned. 
The Euclidean metric is the one that students are familiar with from geometry so it is a good starting point for learning about metric spaces. 
One  important aspect of a metric is the triangle inequality.
Student are already familiar with the triangle inequality in standard Euclidean metric from their geometry.
That helps them understanding the significance of this inequality in other metrics.  
A: Euclidean distance can be seen as the natural distance we encounter in our daily life. This is because Euclidean distance remains invariant under rotation, as the distance of objects is in real life. I am not much of an historian, but if I would need to measure things without proper equipment one of the first things I would do is placing them parallel to each other to be able to compare them. (i.e. rotating the vectors) 
What Pythagoras theorem shows is that this concept of distance we have in the natural world satisfies the equation $x^{2}+y^{2}=z^{2}$. So if, as a mathematician, we want to look at vector spaces modelling the real world it makes sense to use the Euclidean distance. As Mohammad Riazi-Kermani noted in his answer this is also one of the reasons students get introduced to this metric first. A lot of mathematical concepts were inspired by the real world, not the other way around.
