To answer the questions, we need to review the properties of the diagonals of a quadrilaterals.
For parallelograms, the diagonals bisect each other.
If the parallelogram is a rectangle, then the diagonals are also congruent.
If the parallelogram is a rhombus, on the other hand, then the diagonals are perpendicular. It means that they form right angles at their intersection.
If the quadrilateral is a square, then its diagonals bisect each other, are congruent and are perpendicular as well because a square is a parallelogram, a rectangle, and a rhombus all at the same time.
If the quadrilateral, on the other hand, is a trapezoid, there is only a special relationship between the diagonals if it is an isosceles trapezoid (i.e. it has two congruent non-parallel sides)--the diagonals are congruent.
If the quadrilateral is a kite, then the diagonals do not bisect each other, but are perpendicular.
Let's go back to the questions.
The first one tests if the midpoints are the same. This means that the diagonals bisect each other because both of them pass through the same midpoint. Then, it is a parallelogram.
The second one tests if the slopes of the diagonals are negative reciprocals which means that they are perpendicular to one another. This is the case for either a rhombus or a kite.
The third one tests the distance of the diagonals and if they are congruent. This is for an isosceles trapezoid.
If you're testing the slope, midpoint, and distance, then you're testing if the quadrilateral is a square.
The next one does not test diagonals but rather the sides. If only 2 of the sides are parallel, the nit is a trapezoid by definition.
Finally, if none of the tests hold true, then we simply have an irregular quadrilateral.