to sag / drop
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to sag / slump
To move slowly downward or collapse inward, usually because of weight, tiredness, or weakening support.
Nach der langen Wanderung sackte er erschöpft in den Sessel.
After the long hike, he slumped exhausted into the armchair.
The verb 'sacken' (to sag/drop) describes a change of state or position. Therefore, the perfect tense is always formed with the auxiliary verb 'sein' (to be).
Die Kiste ist plötzlich vom Regal gesackt.
The box suddenly slipped down from the shelf.
When a person 'sackt' (sags/slumps), a preposition with the accusative is often used to indicate direction. Typical phrases are 'in den Stuhl sacken' (to slump into the chair) or 'auf die Knie sacken' (to sink to one's knees).
Als er die gute Nachricht hörte, sackte er erleichtert auf den Boden.
When he heard the good news, he sank to the floor in relief.
In the meaning of 'fallen' (to fall), 'sacken' is often used for abstract things such as prices, rates, temperatures, or even the mood.
Als die Sonne unterging, sind die Temperaturen schnell gesackt.
As the sun went down, the temperatures dropped quickly.
The expression 'etwas sacken lassen' (to let something sink in) has a fixed structure. The verb 'lassen' (to let) is conjugated, while 'sacken' remains in the infinitive at the end of the sentence. The thing being processed is in the accusative.
Er hat mir ein kompliziertes Angebot gemacht. Das muss ich erst mal sacken lassen.
He made me a complicated offer. I need to let that sink in first.
to fall
'fallen' and 'sacken' can both describe a downward movement or a decline. 'fallen' is the normal, general word: a glass falls to the floor, prices fall, or the temperature falls. 'sacken' is used more when something slowly slides downward or slumps in on itself, for example a body from tiredness. With prices or rates, 'sacken' often sounds a bit more figurative and sometimes stronger. The words can be confused because both work with numbers and levels. For neutral statements, 'fallen' is usually safer; 'sacken' fits when you want to emphasize giving way or a clear slide downward.
Nach der schlechten Nachricht sackten die Kurse an der Börse deutlich.
After the bad news, stock market prices dropped sharply.
Im Winter fällt die Temperatur hier oft unter null Grad.
In winter, the temperature here often falls below zero degrees.