Because it's a right-handed batter, that is a fair ball.
Seriously, though, what you describe is a fair ball, no matter if the batter is right or left handed.
A bounding ball (one that has bounced) in fair territory that goes directly over first or third base, and then lands in foul territory is a fair ball.
It is a fair ball when any portion of the ball bounds over the front edge of first/third base, where the base meets the foul/fair edge of the foul line. If the ball bounds into foul territory before passing these points, that would be a foul ball.
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