if it was 'il piatto e tuo' it would be okay to not put the article 'il' because the possessive is on the end of the sentence, but since after the possessive 'tuo' there is the subject 'piatto', then you must use 'il' before 'tuo'. I hope you to understand this (I'm an Portuguese native speaker)
My lesson wanted a translation, "The plate is your dish." Very strange--using two translations of the same word. I think this sentence should be amended further. Natural English translation would be The plate is yours." Though one could not say that in Italian while the article is in front of the possessive.
Under "Il gatto é mio" we were told we did not need to repeat "Il" before "Mio". Why do we need to repeat the "Il" in this sentence? Is it because "Piatto" is repeated? Am I correct in thinking that "Il gatto é il mio gatto" and "Il piatto é tuo" would be grammatically correct, if not direct translations of the Italian?