Question

given point a is not on line m, suppose a is reflected across line m. which of the following is true?
○ \\(\overline{aa}\\) is the bisector of line m.
○ line m is the perpendicular bisector of \\(\overline{aa}\\)
○ line m is parallel to \\(\overline{aa}\\)
○ \\(a = a\\)

Explanation:

1. Recall the definition of a reflection across a line: When a point \( A \) is reflected across line \( m \) to get \( A' \), line \( m \) is the perpendicular bisector of the segment joining \( A \) and \( A' \) (i.e., \( \overline{AA'} \)).
2. Analyze each option:

• Option 1: A bisector of a line (which is infinite) doesn't make sense in this context, and \( \overline{AA'} \) is a segment, not a bisector of line \( m \).
• Option 2: By the definition of reflection, line \( m \) is perpendicular to \( \overline{AA'} \) and bisects it (since the distance from \( A \) to \( m \) equals the distance from \( A' \) to \( m \)).
• Option 3: Line \( m \) is perpendicular to \( \overline{AA'} \), not parallel.
• Option 4: Since \( A \) is not on line \( m \), \( A \) and \( A' \) are distinct, so \( A

eq A' \).

Answer:

Line \( m \) is the perpendicular bisector of \( \overline{AA'} \) (the second option).