- this would be "Du spielst rechtshändig" for example tennis
- "Du spielst rechts." is rather a short form - for example in soccer - to talk with your team - ... and partition the work ... "hey, du spielst links - ich spiele rechts - klar" ... this means "you play on the left side of the football ground and i play on the right side."
Not an expert but I've heard phrases like "Du hast recht" used in conversation to say someone is correct. I believe "rechts" with an s is the genitive form of the word and so can only be used for the "right side" as that's the only definition that can reasonably "possess" something. It makes sense to say "the right side (of the political spectrum, the sports field, etc...) has a thing", but not "the correctness has a thing".
TL;DR: recht in some forms can be used to assert correctness, just not the genitive.
All that said, I'm just another learner like you, so I don't guarantee "ich habe recht"!
It does make sense. If you are playing a game with friends... like baseball or basketball... then you could tell your friend where to play. "Hey! You play on the right. I'll play on the left." You have to remember that not all sentences make sense. And I am sure that is worldwide.
Also, since we are learning, maybe the people making DuoLingo didn't want too many words in the sentence. People seem to become frustrated when there are more than five new words to remember in a sentence. You should try picking five words out of the dictionary that you have never heard and make a sentence with them. Chances are that you won't remember them if you don't practice. The people at DuoLingo have done a wonderful job teaching. I only done five lessons, then quit learning over the holidays, but I have retained what everything means and only get confused with remembering how to spell the words. Just keep practicing and have fun. If you don't have fun, then it will be harder to learn. Remember to take a break at least once or twice an hour. Stretch and trinkt das Wasser!
In football/soccer, which this phrase is probably being taken from, a person playing 'on the right' will sometimes be on the left-side of the field.
Say you're a defender playing 'on the right' you would be on the right-side of the field most of the time, but sometimes, if the person you're marking moves to the left for example and you have to follow them, you'll end up on the left-side.
It's basically a description of the responsibility a player has, it covers what area they have to mark and which players they have to mark and so that's why a player 'on the right' can sometimes actually be on the left.
It's not logical, sports idioms never are.