ruhen
to rest
To rest, not be active, in order to recover.
Nach der langen Wanderung müssen wir erst einmal ruhen.
After the long hike, we first need to rest.

to rest
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to rest
To rest, not be active, in order to recover.
Nach der langen Wanderung müssen wir erst einmal ruhen.
After the long hike, we first need to rest.

'Ruhen' is the verb corresponding to the noun 'die Ruhe' (rest/quiet). When something 'ruht' (rests), it is in a state of rest. This connection helps you understand the different meanings of the verb.
Am Sonntag möchte ich nur meine Ruhe haben und auf dem Sofa ruhen.
On Sunday I just want to have my peace and rest on the sofa.
Although 'ruhen' describes a state, it forms the perfect tense with the auxiliary verb 'haben' (to have), not with 'sein' (to be). So the correct form is 'hat geruht' (has rested).
Der Patient hat nach der Operation lange geruht.
The patient rested for a long time after the operation.
When an activity is temporarily suspended, 'ruhen' is often combined with abstract nouns such as 'die Arbeit' (the work), 'der Verkehr' (traffic), or 'die Bauarbeiten' (construction work).
Wegen des Streiks ruht der gesamte Bahnverkehr.
Because of the strike, all rail traffic is at a standstill.
When something lies on a surface or is supported by something, you use 'ruhen auf' (to rest on) with the dative. This applies both literally and figuratively to responsibility or expectations.
Sein Blick ruhte nachdenklich auf dem alten Foto.
His gaze rested thoughtfully on the old photo.
The fixed expression 'etwas ruhen lassen' (to let something rest) means to deliberately not pursue a matter or dispute further. The topic that you are dropping is in the accusative.
Wir kommen hier nicht weiter. Lassen wir das Thema für heute ruhen.
We're not getting anywhere here. Let's let the topic rest for today.
In connection with death, 'ruhen' (to rest) is a very formal and respectful expression for 'begraben sein' (to be buried). You often find it on gravestones, e.g. in the form 'Hier ruht ...' (Here rests ...).
Auf diesem Friedhof ruht ein berühmter Dichter.
A famous poet rests in this cemetery.
to lie / be located
'liegen' and 'ruhen' can both describe a calm, often horizontal position. 'liegen' is the normal everyday word for a body position or the location of an object. 'ruhen' puts more emphasis on rest, recovery, or stillness and often sounds a little more elevated. In 'auf etwas ruhen' (to rest on something), it can also mean that something lies on a surface or is supported by it. If you simply want to say where someone or something is, 'liegen' usually fits. If the rest or stillness is important, 'ruhen' fits better.

Nach der langen Wanderung ruht Paul eine Stunde auf dem Sofa.
After the long hike, Paul rests on the sofa for an hour.
Die Katze liegt in der Sonne auf dem Teppich.
The cat is lying in the sun on the carpet.