Is there a rectangle whose diagonals are mutually perpendicular?

Answer: yes, such rectangles exist. An example is a square.

For the two diagonals of the rectangle to be mutually perpendicular, the following conditions must be met:

1. Both diagonals must bisect the opposite corners between which they are drawn.

2. All sides of the rectangle must be equal to each other, otherwise the execution of step 1 will be impossible.

There is only one kind of rectangle that would satisfy these requirements and at the same time would have 4 right angles at the vertices. Such a figure is a square.