Annoying Trig

Let us draw a triangle $\triangle {ABC}$ with $|\overline {AB}|=3, |\overline {AC}|=6, \angle {BAC}=x$ . Then $|\overline {BC}|=3\sqrt {5-4\cos x}$ . Let us draw a line $\overline {AD}$ perpendicular to $\overline {AC}$ such that $|\overline {AD}|=2$ . Join $\overline {BD}$ . Then $|\overline {BD}|=\sqrt {13-12\sin x}$ . The minimum of $|\overline {BD}|+|\overline {BC}|$ is $|\overline {CD}|=\sqrt {6^2+2^2}=\sqrt {40}$ , so that the required minimum is $\boxed {40}$ .