If we have two pairs of parallel sides (square, rectangle, parallelogram, rhombus) then the diagonals bisect each other.
However, if we only have one pair of parallel sides, as we do in a trapezoid, we don't necessarily have bisecting diagonals.
Equal length sides doesn't imply perpendicular diagonals, though.
This IS the case in a square, where the diagonals are perpendicular, equal, and bisect each other.
It is the case in a rhombus, where the diagonals are perpendicular and bisect each other, but aren't equal to each other.
Everything you need to know on this is in the Geometry text on page 17...