The most useful tool for identifying the areas the Mongols conquered would be a thematic map. A thematic map displays specific data or themes, such as the extent of territorial conquests, which would provide a visual representation of the Mongols' conquered areas. A physical map would show the geographical features of the region, but may not specifically highlight the areas conquered by the Mongols. A line graph and a bar chart are not typically used to show geographic locations but rather for representing quantitative or numerical data.