Anaphase Microtubule Disassembly
I don't think the microtubules shorten. The disassembly of tubulin (dynamic catastrophe) happens from its + end which in this case is protected by the kinetochore proteins. The separation of chromatids, I guess, happens via a dynein (and in some cases kinesin) mediated movement. 

The wikipedia article on kinetochore is quite detailed and you can have a look at it.