Translative Covering a Square with Isosceles Right Triangles

A translative covering the rectangle a×b with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of a×b) is considered. It is shown that any collection of equal triangles homothetic to T with the total area at least 2 permits a translative covering of 1×1; this bound is tight. It is also demonstrated that any collection of positive homothetic copies of T with the total area at least 3 permits a translative covering of 1×1. Moreover, it is proven that if a≥5+334b, then any collection of triangles homothetic to T with the total area at least 12(a+b)2 permits a translative covering of a×b; this bound is tight.