lügen = to lie (tell an untruth) and liegen = to lie (e.g. in a horizontal position or to be located) whereas legen = to lay (e.g. to lay the book on the table). The use of the verb to "lay" to refer to lying (down or being located) as in "liegen" is often made interchangeably, but incorrectly, with the verb to "lie" and there is some confusion between the two in common usage, but I would say that the essential difference is that "lay" is a transitive verb that corresponds more to "legen" (not "liegen") in that it refers to placing or positioning something that is the object of the verb (e.g. lay an egg), whereas "lie" corresponds to "liegen" referring to the positioning of the subject of the verb (e.g. he lies on the bed), not the object.
I just learn so much from these discussions sometimes! Thank you all for your participation. Especially two of you. the person who pointed out that German is a precise language, that's really helpful to remember. Also, the person who taught me that Hocker means stool. That's especially interesting to me because my mother's maiden name is Stonehocker!
Unter is a dual preposition, meaning it can be followed either by the accusative or the dative case. In the particular case of spatial prepositions, the accusative implies movement (think of an answer to the question 'wohin' - 'where to') and the dative implies stillness (think of an answer to the question 'wo' - 'where').
Here the apple simply sits (well, lies) under the chair, so the dative must be used. If the apple was rolling under the chair, you would use the accusative, as in "Der Apfel rollt unter den Stuhl.".
(please correct me anyone if I'm wrong, I'm new at German and English is not my mother tongue).
For anyone having trouble with liegen and legen (or even for anyone who wants to get the English straight, regarding to lie vs. to lay), check out my Tiny Cards at this link and follow me (Liz Groothof Croddy) to practice German! I am a language educator with a background in linguistics, and of course I love Duolingo and use it daily. https://tiny.cards/decks/28WyRHt6/german-1-12-verbs-liegen-und-legen
"The apple is laying under the chair" was not accepted. Shouldn't it be? I see in the comments that it accepts "lying", but it won't accept "laying". I think the difference is just where you come from and how you were brought up, because I hear both here. I personally prefer "laying" though. "Lying" sounds like "lieing"... Is that a word? Anyway, I'd probably understand it either way, but to someone new to English, they could think that the apple was telling lies from under the chair. That apple must be possessed!
An option for an answer was "The apple lies...", this is not correct. "The apple lays..." would be used since the apple, as an inanimate object, cannot be doing the action of its own power.