NL10 TPTK falls on river > what's the thumb of rule here?

[CaptainSubtext]

PokerStars No-Limit Hold'em, $0.10 BB (9 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: 2+2 Forums)

MP3 ($6.30)
CO ($7.95)
Button ($9.50)
SB ($9.90)
BB ($19.95)
UTG ($9.85)
Hero ($10.10)
MP1 ($6.85)
MP2 ($10.40)

Preflop: Hero is UTG+1 with A[img]/images/graemlins/spade.gif[/img], K[img]/images/graemlins/diamond.gif[/img].
<font color="#666666">1 fold</font>, <font color="#CC3333">Hero raises to $0.4</font>, <font color="#666666">3 folds</font>, CO calls $0.40, <font color="#666666">3 folds</font>.

Flop: ($0.95) 3[img]/images/graemlins/heart.gif[/img], 3[img]/images/graemlins/club.gif[/img], 2[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">Hero bets $0.65</font>, CO calls $0.65.

Turn: ($2.25) 9[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(2 players)</font>
Hero checks, CO checks.

River: ($2.25) A[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(2 players)</font>
Hero checks, <font color="#CC3333">CO bets $1.2</font>, Hero???

I've been raising a lot on this table preflop and postflop because of getting good cards preflop and taking them down preflop and catching good flops only to get a badbeat twice.
Thus people are thinking I don't have good cards all of the time.

Here, after a raise preflop and a c-bet which villain calls and after check-check on the turn I suspect villain has medium pair ,Ax or some face cards.

After the ace hits the river, I check because villain will not call a bet from me with medium pair and face cards. Is my way of thinking right or should I just lead out in case he is slowplaying some big hand? And after a riverbet from him should I just flatcall or checkraise. I think he will only bet if he has an ace and I have him crushed, but I also am thinking "Don't go broke with one pair.".