It's really hard to read an over pair bigger than your over pair when the opp has position and isn't raising. He's so passive he could have anything here. Your play looks good but you might get more value in the future (or save yourself money, same thing) when if you find yourself pot commited by the river and are willing to call an all in...but you're not sure you have the best hand, if you check then call all in. Here only a hand that beats you will call. A lot of hands that beat you will check behind (AA might even check behind on the river here, with the possible strait on board and the chance you hold a set). By betting your last 10$ out of position you are essentially giving that money up to a better hand, or taking down the pot when he folds when you have the best hand. If you have the best hand, you might as well just show it down without riskign that 10$ for nothing. If you check and he pushes, you can still call and get the same result as if you push and he calls. At least by checking the river you will save that final bet some of the time.