Question

line m is the perpendicular bisector of \\( \overline { a b } \\). which of the following is not true?

the distance from point a to point d is the same as the distance from point b to point d.
segment ac is congruent to segment bc.
segment bc is the same length as segment dc.
the distance from point b to point e is the same as the distance from point a to point e.

Explanation:

Step1: Recall the property of perpendicular bisector

A perpendicular bisector of a segment has the property that any point on the perpendicular bisector is equidistant from the endpoints of the segment. Since line \(m\) is the perpendicular bisector of \(\overline{AB}\), for any point \(P\) on line \(m\) (such as \(E\) and \(D\)), \(PA = PB\). So, \(AD=BD\) (because \(D\) is on \(m\)) and \(AE = BE\) (because \(E\) is on \(m\)). Also, since \(m\) is the bisector of \(\overline{AB}\), \(AC=BC\).

Step2: Analyze each option

• For the option “The distance from Point \(A\) to Point \(D\) is the same as the distance from Point \(B\) to Point \(D\)”: Since \(D\) is on the perpendicular bisector \(m\) of \(\overline{AB}\), by the perpendicular - bisector theorem, \(AD = BD\).
• For the option “Segment \(AC\) is congruent to Segment \(BC\)”: Since \(m\) is the bisector of \(\overline{AB}\), \(AC = BC\) (by the definition of a bisector).
• For the option “Segment \(BC\) is the same length as Segment \(DC\)”: There is no geometric theorem or property (given that \(m\) is the perpendicular bisector of \(\overline{AB}\)) that would imply \(BC=DC\). \(BC\) is related to the bisecting of \(\overline{AB}\), and \(DC\) has no such relation to the given information about the perpendicular bisector of \(\overline{AB}\).
• For the option “The distance from Point \(B\) to Point \(E\) is the same as the distance from Point \(A\) to Point \(E\)”: Since \(E\) is on the perpendicular bisector \(m\) of \(\overline{AB}\), by the perpendicular - bisector theorem, \(AE=BE\).

Answer:

Segment \(BC\) is the same length as Segment \(DC\).