A classic AA RR hand

[bilbo-san]

It's like me 10th hand at the table. Villain is 19/8/2 over about 50 hands and I have no reads. $200 stacks.

UTG limps, Villain raises to $8, I make it $29 from the BB with A[img]/images/graemlins/club.gif[/img]A[img]/images/graemlins/heart.gif[/img], UTG folds, Villain calls.

Flop is K[img]/images/graemlins/spade.gif[/img]7[img]/images/graemlins/heart.gif[/img]9[img]/images/graemlins/heart.gif[/img]
I check, Villain checks

Turn is the 4[img]/images/graemlins/diamond.gif[/img]
I bet $45, Villain calls.

River is the Q[img]/images/graemlins/spade.gif[/img]

Check or bet?

If we bet, why, and are we folding if he raises, and why?

If we check, why?