Half the distance is used when a penalty has occured and the yardage that is penalized is longer than half the distance to the goal. Example: a team is on their own 10 yard line and they have a 10 yard holding penalty. 10 yards would put them in their own endzone, but they would penalize half the distance and they would start the next drive on their own 5 yard line.
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Anytime the penalties normal yardage (5,10,15 yds) would put the team with the ball in the endzone or outside the field of play then the yardage is changed to 1/2 the distance to the goal. This can make some penalties worth less than 1 yd of penalty but your in a tight spot!
If any part of the football crosses the plane of the goal line it is considered a touchdown.