Question

1. 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. is point e closer to point a, closer to point b, or the same distance between the points? explain how you know.

Explanation:

Step1: Recall perpendicular - bisector property

The perpendicular bisector of a line segment \(AB\) is the set of all points that are equidistant from \(A\) and \(B\). That is, if \(CD\) is the perpendicular bisector of \(AB\), for any point \(P\) on \(CD\), \(PA = PB\), and for any point \(Q\) not on \(CD\), \(QA
eq QB\).

Step2: Analyze the position of point \(E\)

Point \(E\) is said to be closer to point \(A\) and closer to point \(B\). This means that \(E\) is not on the perpendicular bisector \(CD\) of \(AB\). Since points on the perpendicular bisector of \(AB\) are equidistant from \(A\) and \(B\), and \(E\) has unequal distances to \(A\) and \(B\), point \(E\) does not have the same distance between \(A\) and \(B\).

Answer:

No, point \(E\) does not have the same distance between points \(A\) and \(B\) because the perpendicular bisector of \(AB\) is the set of points equidistant from \(A\) and \(B\), and \(E\) is not on the perpendicular bisector as it is closer to one of the endpoints.