Learn Arabic (Modern Standard): رأينا راعيا في الوادي.

راعيا is a 'defective noun' (a noun ending in a weak letter). When these nouns are indefinite and in the nominative or genitive case, they drop the final 'ya' and take a double kasra (like راعٍ). However, because 'shepherd' is the direct object of the verb رأينا (we saw), it is in the accusative case. In the accusative, the 'ya' returns and takes the standard double fatha ending to show it is the object.

Why doesn't الوادي take a visible kasra on its last letter since it comes after the preposition في?

الوادي is indeed in the genitive case because of the preposition في. However, because it is a defective noun ending in a 'ya' (ي), the kasra is considered too heavy to pronounce. Therefore, the vowel is left implicit and the spelling remains الوادي rather than taking a written kasra.

If الوادي means 'the valley', how would we write 'in a valley' to make it indefinite in this sentence?

If it were indefinite ('a valley'), it would still be in the genitive case without the definite article. Following the rule for indefinite defective nouns in the genitive, it would lose its final 'ya' and take a double kasra, becoming في وادٍ.

How does the spelling of the verb رأينا (we saw) work with the hamza and the weak root letter?