Question

select the correct answer. which sentence states a property or a definition used in the construction of a perpendicular bisector? a. a line that intersects a line segment at an angle of 90° is parallel to the line segment. b. a line that bisects a line segment is necessarily perpendicular to the line segment. c. any two points equidistant from the midpoint of a line segment lie on its perpendicular bisector. d. any point on a perpendicular bisector is equidistant from each endpoint of the line segment.

Explanation:

• Option A: A line intersecting a segment at \(90^{\circ}\) is perpendicular, not parallel. So, this option is incorrect.
• Option B: A bisector divides a segment into two equal parts. But a bisector is not necessarily perpendicular. For example, a median of a line segment (a bisector) in a non - isosceles triangle (in a geometric sense) is not perpendicular. So, this option is incorrect.
• Option C: The property is that any point on the perpendicular bisector (not just two points equidistant from the mid - point) is equidistant from the endpoints. The mid - point is equidistant from the endpoints, but the statement in this option is not the key property used in construction.
• Option D: The key property used in the construction of a perpendicular bisector is that any point on the perpendicular bisector of a line segment is equidistant from the two endpoints of the line segment. This property is used to construct the perpendicular bisector (by finding points that are equidistant from the endpoints).

Answer:

D. Any point on a perpendicular bisector is equidistant from each endpoint of the line segment.