Which area had its border set at the 49th parallel by president polk and britain in 1849?

The answer is Oregon. This is solidified through the treaty called the Treaty of Oregon. In this treaty,  the US and the British government had created a boundary to stop states to fight over their territories. The treaty had stated that Oregon county is a joint territory between the US and the British government.

This area was Oregon which was brought about by the Oregon boundary dispute, better known as the Oregon Question. This was an issue regarding the political division of the Pacific Northwest regions located in North America versus several nations that had competitive territorial and commercial aspirations over the area.

Possess private property.