On the 10th of Tebet there are only two prohibitions: eating and drinking. NO additional limitations apply, such as the prohibition of wearing leather shoes, working, driving, washing the body, etc.
Most contemporary Rabbis (R. E. Melamed, Rab O. Yosef) authorize to wash one’s mouth or brush one’s teeth in this Ta’anit, when necessary, provided you will be very careful to lower your head as to avoid swallowing water unintentionally.
This point requires more explanation.
In 1949, and before the day of Yom HaShoa was established, the Chief rabbinate of Israel decided that the Tenth of Tebet should be the national remembrance day for the victims of the Holocaust. They recommended traditional Jewish ways of remembering the dead, such as the study of Mishna Mikvaot, saying Tehilim lighting a candle and a public massive recitation of the Kaddish for those Holocaust victims whose names, and date of death remains unknown. Fasting, the most common Jewish expression of sorrow, was already prescribed for this day.
In Israel many people felt that the horror of the Holocaust should be remembered on its own, and a special day should be dedicated to the Shoa victims’ memory. “For the Holocaust survivors there was only one day worthy of being a memorial anniversary for the Holocaust–April 19, the beginning day of the Warsaw ghetto revolt the greatest revolt of them all, the uprisings that had held the Nazis at bay for a longer period than the great French army” (I. Greenberg). That is how the 27 of Nissan was chosen to commemorate Yom haShoa. Yom HaShoah was inaugurated in 1953, by a law signed by the Prime minister of Israel David ben Gurion.
Since then and in practical terms, we have two public days in which we mourn for the Holocaust: Yom haShoa, the official day, and ‘asara betebet, in which we fast, and say the Kaddish Klaly to remember all those unknown victims of the Nazi genocide.
Who is exempted from fasting?
Minors: boys under 13 and girls under 12 years old are completely exempted from fasting.
Nursing women: According to the Sephardic Minhag, after giving birth women are exempted from fasting for 24 months, even if they are not actually nursing their baby.
Pregnant women, especially after the first 3 months, are exempted from fasting.
A person who feels sick, for example, flu or fever or one who has a chronic disease, for example, diabetes, should not fast.
Elders should consult with their physicians if the fast will not affect their health. If it will, they are exempted (and in some cases, prohibited) from fasting.