blind vs blind confusion

How would you play this? I'm thinking I should've just check-called the river and bet more on the turn.

no good reads really - BB was new to the table, and posted a blind out of turn but has been tight so far.
I had won a few pots pre flop or on the flop since he joined.

Ultimate Bet
No Limit Holdem Ring game
Blinds: $0.50./$1.
6 players
Converter

Stack sizes:
UTG: $109.60
UTG+1: $37.25
CO: $122.20
Button: $60.80
Hero: $109.30
BB: $91.50

Pre-flop: (6 players) Hero is SB with 4 A
4 folds, Hero calls, BB checks.

Flop: T 4 A ($2, 2 players)
Hero bets $2, BB calls quickly.

Turn: J ($6, 2 players)
Hero bets $3.75, BB calls (quickly again).

River: 8 ($13.5, 2 players)
Hero bets $8.5, BB raises to $39, Hero folds.
Uncalled bets: $30.5 returned to BB.

Results:
Final pot: $30.5

benny,

I play it excactly the same - I think I bet more on the turn though (~$5)

Raise PF.
That way he might fold right then and there...if not then you can bet more on the flop and take it down there.

standarrrrd

I play it the same except for raising PF