Question

6 from unit 1, lesson 3 in this diagram, line segment cd is the perpendicular bisector of line segment ab. assume the conjecture that the set of points equidistant from a and b is the perpendicular bisector of ab is true. select all statements that must be true. a am = bm b cm = dm c ea = em d ea < eb e am < ab f am > bm

Explanation:

Step1: Analyze option A

Since \(CD\) is the perpendicular bisector of \(AB\), by the definition of a perpendicular bisector, \(M\) is the mid - point of \(AB\). So, \(AM = BM\).

Step2: Analyze option B

There is no information given in the problem that suggests \(CM=DM\). The perpendicular bisector property is about the relationship between points on the bisector and the endpoints of the segment being bisected (\(A\) and \(B\)), not about the lengths of segments on the bisector itself (\(CM\) and \(DM\)).

Step3: Analyze option C

There is no information given that \(E\) is on the perpendicular bisector in a way that would make \(EA = EM\). The property is that points on the perpendicular bisector of \(AB\) are equidistant from \(A\) and \(B\), not from \(A\) and \(M\).

Step4: Analyze option D

Since \(E\) is not on the perpendicular bisector of \(AB\) (if \(E\) were on the perpendicular bisector \(EA=EB\)), and we have no information to suggest \(EA

Step5: Analyze option E

Since \(M\) is the mid - point of \(AB\), \(AB=AM + BM\) and \(AM>0\), \(BM>0\). So, \(AM=\frac{1}{2}AB\) (because \(AM = BM\)), and \(AM

Step6: Analyze option F

Since \(M\) is the mid - point of \(AB\), \(AM = BM\), so \(AM>BM\) is false.

Answer:

A. \(AM = BM\), E. \(AM