Geometry

$\overline{AE}= \sqrt {\left( \frac {1} {2} \right)^2 + \left( \frac {5\sqrt{3}} {2} \right)^2} = \sqrt {\frac {1} {4} + \frac {75} {4}} = \frac {1} {2} \sqrt {76} = \boxed{\sqrt {19}}.$