When Tenth of Tevet occurs on a Friday, you may not start Shabbat early in order to cut short the fast.  That is, you may not eat until dark, as is normally the case.
 

You may eat or drink as necessary on the fasts of the Tenth of Tevet, 17^th of Tamuz, and Tzom Gedalia if you are sick, faint, or dehydrated, but only enough to resolve your health difficulty.  Consult a rabbi.

All fasts start 72 minutes (normal time) before sunrise except for the Ninth of Av and Yom Kippur.  These include:

• 17th of Tamuz
• 10th of Tevet
• Tzom Gedalia
• Fast of Esther
• Fast by a groom and bride on their wedding day
• Fast for first-born males on the day before Passover. This includes first-born cohanim and Levites (levi'im).

You may brush your teeth on Yom Kippur and all other fast days without water and without toothpaste.  (You may not use water to brush your teeth even on Tenth of Tevet and 17^th of Tamuz.)

Introduction to Fast Days
The purpose of Jewish fast days is to make us reflect on our behavior and improve it by:

• Eliminating negative actions and thoughts, and
• Taking more care in our observance of the commandments.

God told the Jewish people on many occasions that He did not want their pointless sacrifices or fast days but rather wanted the Jews to improve their behavior. If people fast but still have reprehensible behavior, the people have missed the point and purpose of bringing sacrifices and having fast days.
All fast days may be circumvented for health reasons. Consult a rabbi.
Fast days come in several varieties:

• Major fast days: Yom Kippur and the Ninth of Av (Tish'a B'Av). 
• Minor fast days: There are four minor fast days; see below.
• Fast for First-Born
• Individual Fasts

The Major Fasts
Yom Kippur and the Ninth of Av begin before sunset and are both slightly longer than 24 hours. Yom Kippur is a festive day while, on the Ninth of Av, Jews mourn for the destruction of both Jerusalem Temples (the first was destroyed by Babylonians; the second, by the Romans). On both fasts, it is forbidden to:

• Eat or drink.
• Wear leather shoes (but you may wear any other clothing made of leather).
• Bathe (or even to wash anything more than fingers; you may not use a deodorant--not even a spray type--since it is similar to washing).
• Have marital relations.
• Use cosmetics or body scents.

The Minor Fasts
The four minor fasts begin 72 minutes before local sunrise. They technically end at dark but functionally end after ma'ariv (and for the Fast of Esther, after hearing the megila). Unlike most times in the Jewish calendar, this 72 minutes is normal time and NOT based on the current length of the day (sha'a zmanit):

• Fast of Esther (Ta'anit Esther): Day preceding Purim (or sometimes earlier).
• Fast of Gedalia (Tzom Gedalia): Usually the day after Rosh Hashana.
• Tenth of Tevet (Asara b' Tevet):
• Seventeenth of Tamuz (Shiv'asar b'Tamuz): Beginning of the Three Weeks of semi-mourning beforeTish'a b'Av.

On all fast days, eating and drinking are forbidden. The Tenth of Tevet and the Seventeenth of Tamuz additionally have the same restrictions as the Nine Days. So you may wash or bathe on the Fast of Esther and on Tzom Gedalia but not on the Tenth of Tevet and the Seventeenth of Tamuz. (See Nine Days: Restrictions).
Note On a fast day that has been delayed for Shabbat, there may be leniencies for:

• The mohel, sandak, and father of a baby getting a brit mila to eat after mincha.
• Pregnant or nursing women (this leniency applies to eating throughout the day).

Fast for First Born
Fast for first-born males 13 years old and older, on the day before Passover. This includes first-born cohanim and Levites (levi'im). The fast begins 72 normal minutes before sunrise and ends with kiddush at the seder. However, the widespread custom is to attend a siyum on the day before the first seder so that the first-born men do not need to fast on that day.
 
Personal Fasts
Fast by a groom and bride on their wedding day is an example of personal fasts. This fast begins 72 normal minutes before sunrise and ends with drinking wine under the chuppa.