Question

consider the incomplete paragraph proof.
given: p is a point on the perpendicular bisector, l, of mn.
prove: pm = pn
because of the unique line postulate, we can draw unique line segment pm. using the definition of reflection, 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 over the perpendicular bisector \(l\) of \(\overline{MN}\), since \(l\) is the perpendicular bisector, points \(M\) and \(N\) are symmetric with respect to line \(l\). In a reflection over line \(l\), if we consider the reflection of \(\overline{PM}\) over \(l\), and we know that reflections map a point to its symmetric point with respect to the line of reflection. Given that \(l\) is the perpendicular bisector of \(\overline{MN}\), point \(N\) is the reflection (image) of point \(M\) over line \(l\) (because the perpendicular bisector is the set of points equidistant from \(M\) and \(N\), and reflection over the perpendicular bisector swaps \(M\) and \(N\)). The other options: point \(Q\) is on the perpendicular bisector (so its image is itself), and segments \(PM\) and \(QM\) are not points (the blank is asking for the pre - image of a point \(N\) under reflection, so we need a point).

Answer:

point \(M\)