Yes. it is possible to bisect a segment with a perpendicular segment. Follow the link to learn how to do it: http://www.mathopenref.com/constbisectline.html
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No
Perpendicular Bisector
The perpendicular bisector of a line segment AB is the straight line perpendicular to AB through the midpoint of AB.
Biconditional Statement for: Perpendicular Bisector Theorem: A point is equidistant if and only if the point is on the perpendicular bisector of a segment. Converse of the Perpendicular Bisector Theorem: A point is on the perpendicular bisector of the segment if and only if the point is equidistant from the endpoints of a segment.
In general they are not; they are perpendicular only if the rectangle is also a sqare. However, the diagonals of a retangle have another attribute: They are of equal length and bisect each other.