Question

given point a is not on line m, suppose a is reflected across line m. which of the following is true? a aa is the bisector of line m. b line m is the perpendicular bisector of aa. c line m is parallel to aa. d a = a.

Explanation:

Step1: Recall reflection property

When a point \(A\) is reflected across a line \(m\) to get \(A'\), the line \(m\) is the perpendicular bisector of the line - segment \(\overline{AA'}\). This is a fundamental property of reflection in geometry.

1. Option A: \(\overline{AA'}\) cannot be the bisector of line \(m\). A line - segment cannot bisect a line in the context of reflection.
2. Option B: By the definition of reflection, line \(m\) is the perpendicular bisector of \(\overline{AA'}\). When we reflect a point \(A\) across a line \(m\) to get its image \(A'\), the shortest distance from \(A\) to \(A'\) is such that \(m\) cuts \(\overline{AA'}\) at a right - angle and divides \(\overline{AA'}\) into two equal parts.
3. Option C: If line \(m\) is parallel to \(\overline{AA'}\), then \(A\) cannot be the reflection of \(A'\) across \(m\) because the perpendicular distance from \(A\) to \(m\) and \(A'\) to \(m\) would not be equal in the way required for reflection.
4. Option D: \(A\) and \(A'\) are distinct points (since \(A\) is not on \(m\)), so \(A

eq A'\).

Answer:

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