cut texture with cut object

I have an application where I cut an object, but I also want to cut the texture in half. How could I do that. Is there some way to map only a portion of a texture to an object. For instance the object is a plane and I cut it in half but at an angle. I want to now map two new textures to the new pieces.

The project is a microscopy project where the textures are jpeg cross sections of mitochondria mapped onto planes.

thanks for any tips. I've been looking at getPixels but don't understand how to choose an area using it, especially one shaped like a triangle?

Dan