Final answer:
If O is the center of the circle and PX and XQ are segments of a chord PQ bisected by a perpendicular from O, then PX equals XQ.
Step-by-step explanation:
If O is the centre of the circle and PQ is a chord, the statement 'O is parallel to PQ' doesn't accurately apply since O is a point and cannot be parallel to a line. However, assuming the intention was to state that a line or radius from O is perpendicular to PQ, then the relation between PX and XQ would be that PX equals XQ if X is the midpoint of the chord PQ.
This is because in a circle, any perpendicular drawn from the center of the circle to a chord will bisect the chord into two equal segments. Hence, if point X lies on the chord PQ and is the point where the perpendicular from O meets PQ, then PX and XQ are equal in length.
This property is based on the fact that the center of the circle is equidistant from any point on the perimeter of the circle, meaning that radii to the endpoints of the chord create isosceles triangles. The perpendicular bisector of the chord is also a line of symmetry for these triangles, ensuring the two segments of the chord are of equal length.