Rectangle $ABCD$ has $AB=12$ and $BC=5$. Point $M$ is the midpoint of diagonal $\overline{AC}$, and $E$ is on $AB$ with $\overline{ME}\perp\overline{AC}$. What is the area of $\triangle AME$?

$AC=\sqrt{12^2+5^2}=13$, so $AM=\frac{13}2$. By $\triangle AME\sim \triangle ABC$ (right angles at $M$ and $B$, common angle $A$), $AM:AB=13:24$, so $[AME]:[ABC]=169:576$.
$[ABC]=\frac12(12)(5)=30$, so $[AME]=\frac{169}{576}\cdot 30=\boxed{\frac{845}{96}}$.