No, Yes, Yes, No, Yes.No: If the diagonals bisect (in the middle), then the opposite triangles are congruent, and we must have a parallelogram. Yes: Of course, for example like the shape of a kite. Yes: Of course, if all sides are equal and the diagonals also, we have a square. No: A trapezoid is a quadrilateral with two sides that are parallel. The median is the segment that joints the middle of the two non parallel sides. It is parallel to the base, not perpendicular. Yes: It is isosceles. I provide the explanation below. Draw the two perpendiculars from the two top vertices to the base. Clearly, the resulting two triangle rectangles that have the diagonals as hypotenuse are congruent. Let L be the length of the bottom sides of these triangle rectangles. Let K be the length of the bottom side of the trapezoid. The two triangle rectangles that resp. have the left and right sides of the trapezoid as hypotenuse are also congruent. This is because their resp. bottom sides have both the same length  K  L  .
