Final answer:
The Chinese and Aztec empires had relationships with their peripheral states characterized by cooperative alliances and tribute systems. Both empires used these systems for trade, military conquest, political stability, and economic prosperity, involving a complex interdependence and hierarchical structure.
Step-by-step explanation:
The relationship that the Chinese and Aztec empires had with their respective peripheral states during the fifteenth century can be best described as A. Cooperative alliances and tribute systems. This option is most reflective of the historical data that suggests regular interaction between these empires and their neighboring states that involved political, military, and economic coordination, as well as the payment of tribute as a form of subordination and respect.
In the Chinese empire, a distinction was made between the overland system and the tribute system, each handling relations with different neighboring states based on the level of threat they posed. The tribute system, especially, was characterized by ritual submission to the imperial throne, but it still facilitated trade and political recognition in a manner that suggests cooperative alliances rather than domination. Meanwhile, the Aztec Empire, formed through the triple alliance of city-states, utilized cooperative alliances to wage wars of conquest and ensure political stability and economic prosperity through tribute from conquered city-states.
Therefore, both the Chinese and the Aztec empires interacted with their surrounding states through systems that combined cooperation and alliance formation with the collection of tribute, which indicates mutual dependence and structured hierarchy.