I have a right-angled isosceles triangle ABC made out of paper. The right angle is at A. I fold corner B of the triangle over so that it touches the mid-point of the opposite side AC, and press the paper flat to obtain a right-angled quadrilateral.

What is the exact ratio of the area of the new quadrilateral to the area of the original triangle?

(You may find it helpful to cut out your own triangle and fold it to make the quadrilateral.)