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 { a b } \\) have in order for calleys claim to be true? the bisector must be the same length as \\( \overline { a b } \\). the bisector must be perpendicular to \\( \overline { a b } \\). the bisector must intersect but not cross \\( \overline { a b } \\). the bisector must contain a finite number of points.

Explanation:

Step1: Recall the perpendicular bisector theorem

The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
In this case, line segment \(AB\) has mid - point \(M\). For any point on the bisector of \(AB\) to be equidistant from \(A\) and \(B\), the bisector must be the perpendicular bisector of \(AB\). That is, the bisector must be perpendicular to \(AB\).

Step2: Analyze other options

• Option A: The length of the bisector has no relation to the property of a point on the bisector being equidistant from \(A\) and \(B\).
• Option C: A bisector of a line segment (which is a line) will cross the segment \(AB\) at its mid - point \(M\). The statement “intersect but not cross” is incorrect for a line bisector.
• 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.

Answer:

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