Question

1. if overrightarrow{be} is a perpendicular bisector of overline{ad}, which statement is true?

overline{ae}congoverline{ab}

overline{ce}congoverline{bc}

overline{ac}congoverline{cd}

overline{bd}congoverline{ed}

Explanation:

Step1: Recall the definition of perpendicular bisector

A perpendicular bisector of a segment divides the segment into two equal parts. If \( BE\) is the perpendicular bisector of \( AD\), then \( AC = CD\) (by the definition of a bisector: a bisector of a segment splits the segment into two congruent segments).

Step2: Analyze other options

• For \(\overline{AE}\cong\overline{AB}\): There is no information given to suggest that \( AE\) and \( AB\) are congruent. Just because \( BE\) is a perpendicular bisector of \( AD\) does not imply this.
• For \(\overline{CE}\cong\overline{BC}\): There is no geometric principle (such as congruent triangles, mid - point properties related to the given perpendicular bisector condition) that would make \( CE\) and \( BC\) congruent.
• For \(\overline{BD}\cong\overline{ED}\): There is no evidence (e.g., right - triangle congruence (HL - Hypotenuse - Leg) since we don't know about the lengths of other sides related to triangles \( EBD\) and \( ABD\)) to support this congruence.

Answer:

\(\overline{AC}\cong\overline{CD}\) (the third option)