How to rotate and snap an object into a desired position? [Illustrator]

so I'm working with Adobe Illustrator, and I'm trying to have a rectangle pass through the center, and also snap to two points.

Basically, I want the marked (with the black lines) corner of the purple rectangle to be snapped to the top left corner of the right red rectangle. I also want the center of the purple rectangle to pass through the center(as shown by guides). I also want to preserve the width of the purple rectangle. I can easily achieve what I am trying to do by selecting anchor points and snapping them, but I lose the width of the purple rectangle.

I've used Illustrator quite a lot, but I'm always stumped by things like this. How could I do this? There must be some tool or method I don't know about.

Thank you.