jmc

By the Triangle Inequality, $$|z-12|+|z-5i|=|z-12|+|5i-z|\ge |(z-12)+(5i-z)|=|-12+5i|=13.$$But we are told that $|z-12|+|z-5i|=13.$ The only way that equality can occur is if $z$ lies on the line segment connecting 12 and $5i$ in the complex plane.



We want to minimize $|z|$. We see that $|z|$ is minimized when $z$ coincides with the projection of the origin onto the line segment.

The area of the triangle with vertices 0, 12, and $5i$ is $$\frac{1}{2}\cdot 12\cdot 5=30.$$This area is also $$\frac{1}{2}\cdot 13\cdot h=\frac{13h}{2},$$so $h=\boxed{\frac{60}{13}}.$