Question

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 the property of perpendicular bisector

The set of points equidistant from two points \(A\) and \(B\) is the perpendicular bisector of the line - segment \(AB\).

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

Point \(E\) is not on the perpendicular bisector \(CD\) of \(AB\). Since \(E\) is on one - side of the perpendicular bisector \(CD\) of \(AB\), assume \(E\) is on the side closer to \(A\) (without loss of generality). The distance from \(E\) to \(A\) is shorter than the distance from \(E\) to \(B\).

Answer:

If point \(E\) is on the side of the perpendicular bisector \(CD\) of \(AB\) closer to \(A\), then \(E\) is closer to \(A\). If it is on the side closer to \(B\), then \(E\) is closer to \(B\). It is not the same distance between the points because \(E\) is not on the perpendicular bisector of \(AB\).