Question

read the statement and the argument
statement: if point p lies on the perpendicular bisector of a segment ab, then \\( \overline { p a } = \overline { p b } \\)
argument: points on any bisector of a line segment are equidistant from the segments endpoints
decide whether the given statement is true or false and judge the validity of the argument
the statement is true, and the argument is a valid argument to justify the statement
the statement is true, but the argument is not a valid argument to justify the statement
the statement is false because no point on the perpendicular bisector of a line segment is equidistant from the segments endpoints
the statement is false because not all points on the perpendicular bisector of a line segment are equidistant from the segments endpoints

Explanation:

1. Recall the Perpendicular Bisector Theorem: Any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints. So the statement "If point \( P \) lies on the perpendicular bisector of a segment \( AB \), then \( \overline{PA}=\overline{PB} \)" is true.
2. The argument says "Points on any bisector of a line segment are equidistant from the segment’s endpoints". However, a bisector (not necessarily perpendicular) does not guarantee equidistance. But the statement is about the perpendicular bisector. The argument is incorrect in its general claim about "any bisector", but the statement itself is true. Wait, no—wait, the Perpendicular Bisector Theorem is a specific case. Wait, actually, the correct reasoning: The statement is true (by Perpendicular Bisector Theorem). The argument says "any bisector"—but a bisector that is not perpendicular does not have points equidistant from endpoints. However, the statement is about the perpendicular bisector. But the argument's claim is wrong (since not all bisectors have equidistant points), but the statement is true. Wait, no—let's re - examine. The Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints. So the statement is true. The argument says "points on any bisector...", which is false (a non - perpendicular bisector does not have this property). But the statement is about the perpendicular bisector. So the statement is true, but the argument is not a valid argument to justify the statement (because the argument makes a false general claim about any bisector, while the statement is about the perpendicular bisector). Wait, no—maybe I messed up. Let's think again. The Perpendicular Bisector Theorem: Point on perpendicular bisector ⇒ equidistant from endpoints. So the statement is true. The argument says "any bisector"—but a bisector is just a line that divides the segment into two equal parts. A non - perpendicular bisector (e.g., a bisector that is at an angle) will not have points equidistant from the endpoints. So the argument's premise is wrong. But the statement is true. So the correct option is: The statement is true, but the argument is not a valid argument to justify the statement. Wait, no—wait, maybe I made a mistake. Wait, the Perpendicular Bisector Theorem is a theorem, so the statement is true. The argument says "points on any bisector...", which is false. So the argument is not valid. So the correct option is the second one: "The statement is true, but the argument is not a valid argument to justify the statement."

Answer:

The statement is true, but the argument is not a valid argument to justify the statement.