Question

if a point is equidistant from the endpoints of a segment, then it is:

a. the midpoint of the segment.
b. on the segment.
c. on the angle bisector of the segment.
d. on the perpendicular bisector of the segment.

Explanation:

Analyze the geometric statement

Using the Perpendicular Bisector Theorem Converse knowledge point
The statement describes a point \(P\) that is equidistant from the endpoints \(A\) and \(B\) of a segment \(AB\), meaning \(PA = PB\).

Evaluate the options

Using the Perpendicular Bisector Theorem Converse knowledge point

• Option A: The midpoint is only one specific point on the segment, but \(P\) can lie anywhere in the plane as long as \(PA = PB\).
• Option B: \(P\) does not have to lie on the segment \(AB\); it can be above or below it.
• Option C: A segment does not have an "angle bisector" because a segment is not an angle.
• Option D: By definition, the set of all points equidistant from the endpoints of a segment is its perpendicular bisector.

Formulate the final conclusion

Using the Perpendicular Bisector Theorem Converse knowledge point
Therefore, the point must lie on the perpendicular bisector of the segment.

Answer:

• A. the midpoint of the segment.
• B. on the segment.
• C. on the angle bisector of the segment.
• D. on the perpendicular bisector of the segment. (Correct answer)