Question

let m be a line in the plane. choose all the statements that should be in the definition of a reflection t across line m. a. for any point p on line m, t(p)=p. b. for any point p on line m, t(p)=p. c. for any point p not on line m, t(p) is a point p such that (overline{pp}) is the bisector of line m. d. for any point p not on line m, t(p) is a point p such that line m is the perpendicular bisector of (overline{pp}). e. for any point p not on line m, t(p) is a point p such that (overline{pp}) is the perpendicular bisector of line m.

Explanation:

1. For points on the line of reflection: In a reflection across a line \(m\), points on the line \(m\) are invariant under the reflection transformation. That is, if \(P\) lies on \(m\), then the image of \(P\) under the reflection \(T\), \(T(P)=P\). So statement A is correct.
2. For points not on the line of reflection: For a point \(P\) not on the line \(m\), its image \(P'\) under the reflection \(T\) is such that the line \(m\) is the perpendicular - bisector of the line segment \(\overline{PP'}\). This is the fundamental property of a reflection. So statement D is correct.

Answer:

A. For any point \(P\) on line \(m\), \(T(P)=P\)
D. For any point \(P\) not on line \(m\), \(T(P)\) is a point \(P'\) such that line \(m\) is the perpendicular bisector of \(\overline{PP'}\)