Question

logical reasoning tell whether the information in the diagram allows you to conclude that c is on the perpendicular bisector of ab. explain your reasoning. 10.

Explanation:

Step1: Recall the definition of perpendicular bisector

A perpendicular bisector of a segment is a line that is perpendicular to the segment and passes through its mid - point.

Step2: Analyze the given information

In the diagram, since \(CP\perp AB\) (shown by the right - angle symbol) and \(AP = PB\) (shown by the tick marks), \(CP\) is the perpendicular bisector of \(AB\).
By the converse of the perpendicular bisector theorem, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. Here, point \(C\) is on the line \(CP\) which is the perpendicular bisector of \(AB\).

Answer:

Yes, \(C\) is on the perpendicular bisector of \(\overline{AB}\) because \(CP\) is perpendicular to \(AB\) (\(CP\perp AB\)) and \(P\) is the mid - point of \(AB\) (\(AP = PB\)). By the definition of a perpendicular bisector (a line that is perpendicular to a segment and passes through its mid - point), \(CP\) is the perpendicular bisector of \(AB\), and \(C\) lies on \(CP\).