jmc

First of all, a sketch might be useful: It is clear that we have a $3:4:5$ right triangle on our hands, so we have a right angle at $B.$ That means if we can find $DB,$ we can use the Pythagorean Theorem on $△ABD$ to find $A{D}^2.$

To find $DB,$ we see that $DB:DC=AB:AC=6:10$ thanks to the Angle Bisector Theorem. That means $DB:DC:BC=6:10:16,$ since $BC=DB+DC.$ So $DB=\frac{6}{16}\cdot BC=3.$

Finally, we have that $A{D}^2=A{B}^2+B{D}^2=6^2+3^2=36+9=\boxed{45}.$