
sich rächen
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.
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.

to come back to haunt
Describes a situation in which a past bad action or negligence later has negative consequences for the person responsible.
Seine Faulheit rächte sich in der Prüfung.
His laziness came back to haunt him in the exam.

to avenge
To take revenge on behalf of another person or for a specific act.
Der Held wollte den Tod seines Freundes rächen.
The hero wanted to avenge his friend's death.
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.