What is the significance of the Period of a Steam Cipher If we only know the period of a Stream Cipher, let's say 150-200 bit in length, how can we use this information to attack a stream cipher? 

Taking a look at this link, What is the period of a cipher? I see that the period of the stream cipher is simply the amount of data we encrypt before starting the encryption over again. 

From the attacker's stand point, this let's us know the chunk size of data we are encrypting but how does this help us attack the stream cipher? Won't we still need to know plaintext and cipher text?
If you know the period of the stream cipher, you can XOR 2 different encryptions from the same point in the period (integral number of periods apart) and eliminate the pseudo random stream entirely.  All that would be left are 2 plaintexts XOR'd together, and this is something a smart child can break in a short time.