The greatest possible area of a triangle with one side of length 7 and another side of length 10 can be found using the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, the given sides 7 and 10 can be the base and height of the triangle, but we need to determine which side should be the base and which should be the height to maximize the area. The area will be maximized when the longer side (10) is the base and the shorter side (7) is the height, as this creates a taller triangle.

So, the area of the triangle is:

Area = (1/2) * 10 * 7 = 35 square units

Therefore, the greatest possible area of the triangle is 35 square units.