Covering the Triangle

If you begin with an isosceles right triangle (you know, a 45-45-90), it’s very easy to cover it completely with two congruent shapes, or with four congruent shapes:

Much more interesting, and star-worthy, is the task of covering this triangle with three congruent shapes. You may use any shape you want (not just triangles), but all three shapes must be congruent, and they may not overlap, nor extend beyond the edges of the triangle. Complete coverage appears to be impossible, so the question is: what shape and what arrangement will give you the maximum coverage? Also see if you can calculate exactly how much of the triangle your solution covers.