Question

line segment ab is drawn with midpoint m. a bisector is drawn through point m. calley claims that any point on the bisector drawn through point m is equidistant from a and b. which additional property must the bisector of \\( \overline{ab} \\) have in order for calleys claim to be true? a the bisector must be the same length as \\( \overline{ab} \\). b the bisector must be perpendicular to \\( \overline{ab} \\). c the bisector must intersect but not cross \\( \overline{ab} \\). d the bisector must contain a finite number of points.

Explanation:

The perpendicular bisector theorem states that any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment. Since \(M\) is the mid - point of \(AB\), if the bisector is perpendicular to \(AB\), then by the perpendicular bisector theorem, any point on this bisector will be equidistant from \(A\) and \(B\).

• Option A: The length of the bisector has no relation to the equidistant property from \(A\) and \(B\).
• Option C: A bisector of a line segment (which is a line) will always cross the line segment (in this case \(AB\)) at its mid - point \(M\). The description “intersect but not cross” is incorrect for a line bisector of a line segment.
• Option D: A bisector (a line) contains an infinite number of points, and the number of points on the bisector is not relevant to the equidistant property from \(A\) and \(B\).

Answer:

B. The bisector must be perpendicular to \(\overline{AB}\)