the number of triangles having a fixe4d base of "a"units and area 2009square units is

Answer 1

there are infinitely many such triangles.
any number h such that ah/2 = 2009 will work.

However, if a is an integer, and you also want the height h to be an integer, then there is a limit.

4018 = 2* 7^2 * 41, then the only possible values of a and h are

1 4018
2 2009
7 574
14 287
41 98
49 82