To determine the type of transformation described by the given mapping, let's carefully analyze the properties provided.
1. Property 1: Every point [tex]\(A\)[/tex] on line [tex]\(\ell\)[/tex] maps to itself.
This means that the line [tex]\(\ell\)[/tex] is invariant under the transformation. In other words, any point lying on line [tex]\(\ell\)[/tex] does not change its position after the transformation.
2. Property 2: Every point [tex]\(P\)[/tex] that isn't on [tex]\(\ell\)[/tex] maps to a point [tex]\(P'\)[/tex] such that [tex]\(\overline{PP'}\)[/tex] is the diameter of a circle centered on [tex]\(\ell\)[/tex], and [tex]\(\overline{PP'}\)[/tex] is perpendicular to [tex]\(\ell\)[/tex].
This implies that for any point [tex]\(P\)[/tex] not on [tex]\(\ell\)[/tex], the point [tex]\(P'\)[/tex], to which [tex]\(P\)[/tex] maps, must satisfy the following conditions:
- The segment [tex]\(\overline{PP'}\)[/tex] is the diameter of a circle whose center lies on the line [tex]\(\ell\)[/tex].
- The segment [tex]\(\overline{PP'}\)[/tex] is perpendicular to [tex]\(\ell\)[/tex].
Since [tex]\(P\)[/tex] and [tex]\(P'\)[/tex] form the diameter of the circle, the midpoint [tex]\(M\)[/tex] of segment [tex]\(\overline{PP'}\)[/tex] must lie on line [tex]\(\ell\)[/tex]. This indicates that [tex]\(P\)[/tex] and [tex]\(P'\)[/tex] are symmetric with respect to [tex]\(M\)[/tex], and since [tex]\(M\)[/tex] is on [tex]\(\ell\)[/tex], they are symmetric with respect to line [tex]\(\ell\)[/tex].
Considering these properties, we can see that the transformation being described maps each point to its mirror image across line [tex]\(\ell\)[/tex]. This description matches the definition of a reflection across a line.
Therefore, the transformation in question is a reflection. The correct answer is:
(B) A reflection
