Consider the $1,1,\sqrt2$ right isosceles triangle. The desired heptagon is a $19\times 53$ rectangle, with a right isosceles triangle cut off from each of 3 of its corners. It can be dissected into $2\times 19\times 53-3=2011$ right isosceles triangles.
Consider a rhombus of side length $1004$ composed of 2008 isosceles triangles with side lengths in the ratio $1:1004:1004$. Now attach three more such triangles to 3 of the sides on the rhombus, this is a valid construction.