It would be a fair ball.
Home plate is foul territory unless the ball rolls in front of the plate and stays fair. If the ball bounces off the plate and strikes the batter; it's a dead ball.
Fair ball.
It would be considered a fair ball.
The ball is fair. All bases are fair.
fair because first base is in fair territory
Home Plate is in fair territory.
Since the pitcher's plate, or rubber, is in fair territory, it is a fair ground ball. However, if the ball subsequently goes into foul territory inside 1st or 3rd base before being touched by a player or umpire and comes to rest in foul territory, or is touched while in foul territory, it is a foul ball.
I believe, since home plate is in fair territory, the ball would be fair and the batter would be called out for making contact with a live, fair ball. The equivelent to runner interference with the ball on the base paths. ---------- The question is unclear. If a pitched ball hits home plate, it can then be hit by the batter (former Astros player, Enos Cabell, comes to mind). However, if the batter hits the pitch, and the ball then hits the plate and bounces up into the air, and the batter then hits it a second time, he's out. The rule says if the bat hits a batted ball a second time, the batter is out, which is different the the ball hitting the bat a second time.
It can be either. If the ball is hit into the air and is in foul territory at any time, it counts as a foul ball if it goes into the stands. If the ball is in the air and is in fair territory when it goes into the stands, it's a home run. If the ball is in fair territory and hits the ground, then bounces into the stands in fair territory, it's a ground rule double. If the ball bounces in fair territory but then bounces into the stands in foul territory past the infield, it's also a ground rule double. If the ball bounces in fair territory, but then bounces into the stands before it passes 1st or 3rd base, it's a foul ball.
Because home plate is in fair territory, the ball could settle on top of the plate and it would be a fair ball.
Home plate is considered just as much "fair territory" as any other part of the playing field. If a batted ball comes to rest on home plate without being touched by any player, it is a fair ball. If a catcher grabs the ball, when the only part of the field the ball has touched is home plate, the ball is fair -- just as if it had bounced only within the chalk lines. Thus, if a batter hits the ball downward onto the plate, and the ball then bounces into the air, and the catcher grabs the ball in the air before it hits any other part of the field -- the ball is fair. Since, at this point, the batter must run to first base, the catcher could throw the runner out at first.
Home plate is considered in fair territory. If the ball hits home plate and rolls into foul territory, the ball is foul. If the ball hits home plate and rolls into fair territory, the ball is fair.