Constructing an isosceles right triangle with $\sqrt{2}$ sides

Question

Can you actually have an isosceles right triangle with $45$ degree angles?

Such a triangle has sides of $2$, $\sqrt{2}$, and $\sqrt{2}$. But $\sqrt{2}$ is an irrational number. Can you actually have line segments which are $\sqrt{2}$ in length?

Answer 1

The answer is trivially affirmative, we can easily construct an isosceles right triangle (and consequently it has two $45$ degrees angles).

Proof. $\mathbb{R}$ is a complete set (having a cardinality of $2^{\aleph_0}$) and $\sqrt{2}, 2 \in \mathbb{R}$. Now, we can construct our triangle from the square $A$ of area $2$. Just cut it in half joining any pair of opposite vertices in order to get an isosceles right triangle of unit area. Its cathetes length is $\sqrt{2}$, since the side of the aforementioned square measures $\sqrt{\textit{Area of} A}$, which is $\sqrt{2}$.

Thus, we have shown that if a square of area $2$ exists, then an isosceles right triangle with cathetes of $\sqrt{2}$ exists, but $x^2=2 : x \in \mathbb{R}^+$ has one solution in the given field (i.e., $\mathbb{R}^+$) so that our first statement is proven.