Unit circle/Diagonal/Intersection point/Q and R/Exercise

We consider the rational unit circle

${}E={\left\{(x,y)\in \mathbb {Q} ^{2}\mid x^{2}+y^{2}=1\right\}}\,$

and the line

${}G={\left\{(x,y)\in \mathbb {Q} ^{2}\mid x+y=0\right\}}\,.$

1. Determine the intersection points ${}E\cap G$.
2. What is the answer when we consider the situation not over ${}\mathbb {Q}$, but over the real numbers ${}\mathbb {R}$?
3. Can you understand the circle without understanding the the real numbers?
4. Does there exist a relation with the Intermediate value theorem?

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