The construction method used to construct a perpendicular through a point not on a line could also be used to construct a perpendicular bisector.
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The answer for the question above is TRUE.

Given a line segment with endpoints A and B, we must merely find a point on the perpendicular bisector of the line AB, then use the perpendicular line construction. To find the point on the perpendicular bisector, we must merely find a point that is equidistant from both A and B. We can do this with a compass by choosing a compass length greater than half the length AB, and then making an arc from A above the line AB. Then we make an arc from B having the same length. 

Since we used the same length for each arc, the intersection of the arcs, P, is equidistant from A and B. Now we simply use the perpendicular line construction for line AB and P, which is off the line. This line will be a perpendicular bisector. 

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