smc

From point $P$, create perpendiculars to all four sides, labeling them $a,b,c,d$ starting from going north and continuing clockwise. Label the length $PB$ as $x$: We have $a^2+b^2=5^2$ and $c^2+d^2=3^2$, leading to $a^2+b^2+c^2+d^2=34$. We also have $a^2+d^2=4^2$ and $b^2+c^2=x^2$, leading to $a^2+b^2+c^2+d^2=16+x^2$. Thus, $34=16+x^2$, or $x=\sqrt{18}=3\sqrt{2}$, which is option $\boxed{3\sqrt{2}}$