Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?
$\mathrm{(A)}\ \text{the empty set}\qquad\mathrm{(B)}\ \text{one point}\qquad\mathrm{(C)}\ \text{two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}$
\begin{align*}(x+y)^2&=x^2+y^2\\ x^2 + 2xy + y^2 &= x^2 + y^2\\ 2xy &= 0\end{align*}
Either $x = 0$ or $y = 0$. The [union](https://artofproblemsolving.com/wiki/index.php/Union) of them is 2 lines, and thus the answer is $\mathrm{(C)}$.
[asy] draw((0,-50)--(0,50));draw((-50,0)--(50,0));[/asy]