通过参加活动英语

应该没有through this activity的说法吧。要不你就说因为这次活动怎么怎么样。because of this activity,……或者说在这次活动中。in this acticity（下面这个我不知道能不能这么说，建议你问问你的老师，比较保险）

要翻译整句话吗Through taking part in afte-school activities,such as riding horeses or going camping,teenagers are likely to/have the possibility/chance to broaden their horizons and make more friends.