reagieren auf
to react to
To respond with an action or statement to an action, statement, or event.
Er hat auf meine E-Mail immer noch nicht reagiert.
He still hasn't reacted to my email.

to react
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to react to
To respond with an action or statement to an action, statement, or event.
Er hat auf meine E-Mail immer noch nicht reagiert.
He still hasn't reacted to my email.

The verb 'reagieren' (to react) is almost always used with the preposition 'auf' and the accusative. You should memorize this pattern as a fixed unit: 'reagieren auf etwas/jemanden' (to react to something/someone).
Wie hat sie auf die Nachricht reagiert?
How did she react to the news?
Verbs that end in '-ieren' are regular and form the past participle without 'ge-'. So it is 'hat reagiert' (has reacted), not 'hat gereagiert'. This is a useful rule for many verbs of Latin origin.
Ich habe sofort auf seine Frage reagiert.
I reacted immediately to his question.
The context makes clear which meaning is intended. If it is about a person or a message, it refers to a social reaction. If it is about substances such as pollen or nuts, it refers to a physical reaction. In a scientific context, it refers to a chemical reaction.
Sie reagierte nicht auf seine Entschuldigung.
She did not react to his apology.
To describe the type of reaction, 'reagieren' (to react) is often combined with adverbs. Typical examples are 'schnell' (quickly), 'gelassen' (calmly), 'überrascht' (with surprise), 'wütend' (angrily), or 'gar nicht' (not at all) reagieren.
Er hat sehr gelassen auf die Kritik reagiert.
He reacted very calmly to the criticism.
to answer
'beantworten' and 'reagieren' both have to do with a response to something. Use 'beantworten' when there is a question, a message, a letter, or an inquiry and you give a direct answer. It usually has an accusative object: 'eine Frage beantworten' (to answer a question), 'eine E-Mail beantworten' (to answer an email). 'reagieren' is more general. You can react to words, a situation, behavior, or an event. The reaction does not have to be a clear answer; it can also be an action, a feeling, or silence. Learners confuse these words because both come after something that happened before. When it is about a specific question or message, 'beantworten' fits better. When it is about behavior in a situation, 'reagieren' sounds more natural.

Wir reagieren ruhig, wenn es auf der Reise ein Problem gibt.
We react calmly if there is a problem on the trip.
Am Montag hat er die E-Mail des Kunden beantwortet.
On Monday, he answered the customer's email.