Final answer:
The body of water that borders the Persian homeland is the Persian Gulf, which is connected to the Indian Ocean via the Strait of Hormuz and has been historically significant for trade.
Step-by-step explanation:
The body of water that borders the Persian homeland and is relevant to the historical extent of the Persian Empire is the Persian Gulf. This water body not only provided a maritime boundary for the Persian homeland but also facilitated trade and maritime activities. The Persian Gulf is connected to the Indian Ocean by a narrow passageway known as the Strait of Hormuz, which is a strategic choke point in modern times. It is worth noting that historically the Caspian Sea also bordered the northern reaches of the Persian Empire, but it is the Persian Gulf that is closest to the historical heartland of Persia.