What does it mean to turn the other cheek?

Luke uses the word strike which does not mean to slap. It means to beat...hit with the intent to do harm.

Matthew 5:39
"But I tell you. Do not resist and evil person. If someone strikes you on the right cheek, turn to him the other also."
Matthew uses the word that means to hit with a rod.
It can mean to hit with an open hand or a clenched fist.

Romans 12:17-19
"Do not repay anyone evil for evil. Be careful to do what is right in the eyes of everybody. If it is possible, as far as it depends on you, live at peace with everyone. Do not take revenge, my friends, but leave room for God's wrath, for it is written: 'It is mine to avenge; I will repay,' says the Lord."

This goes against everything we hear and see around us!

How do we deal with it?