"The woman eats the orange."
Translation:Kadın portakalı yer.
It's just an unfortunate and confusing coincidence that ayı ends in ı. A bear is ayı, not ay.
So why not "ayıyı" and "Kadını"? Because any noun that is in the subject position /nominative case can be either "a" or "the". There is no difference unless you add "Bir" (a/an/one). The Tips and Notes section on Articles will explain this.
Adding the ı (or i, u, ü) does not work for subjects, ONLY for specific direct objects (the things being eaten, found, bought, etc). The Tips and Notes for the Accusative Case will explain this.
So that's why it's kadın (subject/nominative case) but she's eating portakalı (specific direct object/accusative case). And in your previous question the ayı (the/a bear) was eating fareyi (the mouse).
IF these sentences were the other way around, and the orange was somehow eating the woman (revenge of the fruit!!), it would be: Portakal kadını yer. And if that mouse was VERY hungry: Fare ayıyı yer. :-)
Bu kadın only means "this woman." Similarly, "O kadın" only means "that woman."
Kadını is in the accusative case. The accusative can only be used for direct objects. It will never, ever appear as a subject in a sentence. It only means the woman as the object of a verb. :)
If you just say kadın as a subject, there is no way to tell if you mean "a" or "the" woman. :)