Question

consider the incomplete paragraph proof.
given: p is a point on the perpendicular bisector, l, of (overline{mn}).
prove: (pm = pn)
because of the unique line postulate, we can draw unique line segment pm. using the definition of reflection, (overline{pm}) can be reflected over line l. by the definition of reflection, point p is the image of itself and point n is the image of ________. because reflections preserve length, (pm = pn).
point m
point q
segment pm
segment qm

Explanation:

When reflecting a point over a perpendicular bisector, the reflection of one endpoint of the segment (in this case, since we are dealing with segment \(MN\) and line \(l\) is its perpendicular bisector) will map to the other endpoint. Since we are reflecting over the perpendicular bisector of \(MN\), and we know that reflections preserve the relationship with respect to the line of reflection. If we consider the reflection of \(PM\) over line \(l\), and \(P\) is on the perpendicular bisector (so it maps to itself), then the other point \(M\) (which is an endpoint of the segment \(MN\) that the perpendicular bisector acts on) must map to \(N\) because of the properties of perpendicular bisectors and reflections.

Answer:

point \(M\)