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:

A reflection across a line maps points on the line to themselves. For points not on the line, the line of reflection is the perpendicular bisector of the segment connecting the original point and its image.

• Option A: Correct, as points on the mirror line stay unchanged.
• Option B: Incorrect, this contradicts the definition (points on m map to themselves, not a different P').
• Option C: Incorrect, it lacks the key geometric relationship defining the reflection.
• Option D: Incorrect, the segment $\overline{PP'}$ does not bisect line m; line m bisects $\overline{PP'}$.
• Option E: Correct, this is the defining property for points not on the mirror line.

Answer:

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