Riding a bicycle or car or any other form of transportation is permissible in itself. The Arab woman during the days of ignorance as well as Islam used to ride camels. However, a woman must abide by Islamic mannerisms when riding a bicycle, such as wearing appropriate Islamic dress and avoiding physical contact with men.
As for the possibility of teenage girls losing their hymen; it is important to examine such possibility. If it remains a rare occur[ance], then Islam has decided that a rule cannot be based upon a rarity. However, if it is likely that the girl will indeed lose her hymen if she rides a bicycle and no measures can prevent her from doing so, then the Muslim girl ought to be stopped from this, so that people do not think ill of her and that she is not accused of what she has not committed. However, if riding a bicycle is an actual need for the girl, for instance to get to her school or important work, etc., then it remains that necessities make prohibitions permissible.