Question

the diagram below shows the construction of a line through point p perpendicular to line m. which statement is demonstrated by this construction? 1) if a line is parallel to a line that is perpendicular to a third line, then the line is also perpendicular to the third line. 2) the set of points equidistant from the endpoints of a line segment is the perpendicular bisector of the segment. 3) two lines are perpendicular if they are equidistant from a given point. 4) two lines are perpendicular if they intersect to form a vertical line.

Explanation:

Step1: Analyze each option

• Option 1: This is about the relationship between parallel lines and perpendicularity to a third line. It is not related to the construction of the perpendicular bisector shown in the diagram.
• Option 2: The construction in the diagram shows arcs drawn from the endpoints of a segment (the horizontal line \(m\) can be considered as a segment for construction purposes). The set of points (the intersection points of the arcs) equidistant from the endpoints of a segment is the perpendicular bisector of the segment.
• Option 3: This is about points equidistant from a given point, which is not what the construction (related to segment endpoints) is demonstrating.
• Option 4: This is about the relationship between two intersecting lines forming a vertical line, which is not relevant to the segment - bisecting construction.

Answer:

1. The set of points equidistant from the endpoints of a line segment is the perpendicular bisector of the segment.