Question

segment \\( \overline { a ^ { prime } b ^ { prime } } \\) is the result of reflecting \\( \overline { a b } \\) across line \\( l \\). select all of the correct statements about the unchanged properties of \\( \overline { a b } \\) and \\( \overline { a ^ { prime } b ^ { prime } } \\). choose all answers that apply: \\( \overline { a b } \\) and \\( \overline { a ^ { prime } b ^ { prime } } \\) have the same lengths. \\( b \\) and \\( b ^ { prime } \\) have the same coordinates. none of the above

Explanation:

Step1: Recall the property of reflection

Reflection is a rigid transformation. Rigid transformations (such as reflection) preserve the lengths of line - segments.
If \(\overline{AB}\) is reflected over line \(l\) to get \(\overline{A'B'}\), then by the definition of a rigid transformation (isometry), \(AB = A'B'\).

Step2: Analyze the coordinates of \(B\) and \(B'\)

When a point is reflected over a line (in this case line \(l\)), unless the point lies on the line of reflection, its coordinates change. Since \(B\) is not on line \(l\) (because \(\overline{AB}\) is being reflected to get \(\overline{A'B'}\) and \(B\) and \(B'\) are distinct non - collinear (in the sense of the pre - image and image) with respect to line \(l\)), \(B\) and \(B'\) do not have the same coordinates.

Answer:

A. \(\overline{AB}\) and \(\overline{A'B'}\) have the same lengths.