Can You Fast a Half Day?

Some years, the fast on the 10th Tevet falls on a Friday. This brings into conflict the fast (which finishes at sundown) and shabbat (which begins before sundown). How can we resolve this contradiction between the mourning of the fall of Jerusalem on the 10th Tevet and rejoicing in our shabbat? Ethan Tucker does a comprehensive overview of the main halakhic sources about this issue in this publication from the Hadar Institute.