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In the complex plane, what is the length of the diagonal of the square with vertices \(4,\ 3+5i,\ -2+4i, \text{ and } -1-i \) ?

\(\begin{array}{|rcll|} \hline && |(3+5i)-(-1-i)| \\ &=& |3+5i+1+i| \\ &=& |4+6i| \\ &=& 2|2+3i| \\ &=& 2\sqrt{2^2+3^2} \\ &=& \mathbf{2\sqrt{13}} \\ \hline \end{array}\)

or

\(\begin{array}{|rcll|} \hline && |(-2+4i)-(4)| \\ &=& |-2+4i -4| \\ &=& |-6+4i| \\ &=& 2|-3+2i| \\ &=& 2\sqrt{(-3)^2+2^2} \\ &=& \mathbf{2\sqrt{13}} \\ \hline \end{array}\)