Barry, if you type in your answer, both "half an hour" and "a half hour" are accepted.
I know not a direct translation but "30 minutes" should be ok and is actually accepted elsewhere on Duolingo
a ¨half hour¨ or half AN ¨hour¨ but not mixed together. ¨You have to wait A half AN hour¨ is like saying ¨You have to buy A big A dog. ¨
Tener always uses que to mean to have to, not a. I don't think Duo is inconsistent with tener que + infinitive.
Here are a few verbs that need a before a following infinitive: ir a, venir a, ayudar a, invitar a, empezar a, comenzar a, enseñar a, aprender a, y otros.
I put "you have to wait half hour" should have been accepted. I have reported.
It has to be "you have to wait half an hour". "to wait half hour" is not correct English.
Nora, I think it would be "half an hour" or "a half hour," not "a half an hour."