to take revenge
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to take revenge
To retaliate for an injustice or insult one has suffered. You deliberately harm someone because that person previously harmed you.
Sie schwor, sich für den Verrat zu rächen.
She swore to take revenge for the betrayal.
The verb 'rächen' (to take revenge / to avenge) is almost always used reflexively: 'sich rächen' (to take revenge). The person taking revenge is the subject of the sentence.
Er wollte sich für die Beleidigung rächen.
He wanted to take revenge for the insult.
The construction follows a fixed pattern: you take revenge 'an jemandem' (on someone, dative) 'für etwas' (for something, accusative).
Sie rächte sich an ihrem Ex-Freund.
She took revenge on her ex-boyfriend.
In this meaning, the subject is not a person but something abstract, such as a mistake, a bad habit, or an action. This 'rächt sich' (comes back to haunt someone) later by having negative consequences.
Dass er nicht für die Prüfung gelernt hatte, rächte sich bitterlich.
The fact that he had not studied for the exam came back to haunt him badly.
The rare non-reflexive form 'rächen' (to avenge) means that you take revenge for another person or for a cause. The object in the accusative is what is being avenged.
Im Film will der Protagonist den Mord an seinem Bruder rächen.
In the film, the protagonist wants to avenge the murder of his brother.
to assign / to forgive / to miss
'vergeben' and 'rächen' are opposites here. When you forgive someone, you no longer hold a mistake against them and do not want to harm them in return. When you take revenge, you want retaliation for a wrong you have suffered and deliberately harm the other person. 'vergeben' tends to end the argument, while 'rächen' continues the conflict. So the difference is: letting go or striking back.
Er will sich nicht rächen, obwohl sein Freund ihn sehr verletzt hat.
He does not want to take revenge, even though his friend hurt him deeply.
Nach dem Streit vergibt sie ihrer Kollegin die harten Worte.
After the argument, she forgives her colleague for the harsh words.