Since the area of the square is 49 square inches, the side length of the square is $ \sqrt{49} = 7 $ square inches. Each triangle formed by the fold is a 45-45-90 triangle whose legs are sides of the square and whose hypotenuse is the fold. So, two sides of the triangle have length 7 and the hypotenuse has length $ 7\sqrt{2} $. Therefore, the perimeter of the triangle is $ 7+7+7\sqrt{2} = \boxed{14+7\sqrt{2}} $.