When constructing a perpendicular bisector of a segment, it is essential that the compass opening be greater than [tex]\(\frac{1}{2}\)[/tex] the length of the segment. Here’s why:
1. Intersection of Arcs:
- When you open the compass to more than half the length of the segment, and draw arcs from both endpoints of the segment, these arcs will intersect at two points. These intersection points are crucial because connecting these points gives you the perpendicular bisector.
2. Ensuring Intersection:
- If the compass opening is exactly [tex]\(\frac{1}{2}\)[/tex] the length of the segment, the arcs will touch but not intersect at two distinct points.
- If the compass opening is less than [tex]\(\frac{1}{2}\)[/tex] the length of the segment, the arcs will not even touch each other. Without intersection points, you cannot accurately determine the line that is equidistant from both endpoints of the segment and perpendicular to it.
3. Correct Construction:
- The perpendicular bisector needs to be correctly constructed to ensure that it is exactly perpendicular (forming a 90-degree angle with the segment) and that it correctly bisects the segment (divides it into two equal halves). By making the compass opening greater than half the segment length, the accuracy and validity of the construction are maintained.
Therefore, a compass opening greater than [tex]\(\frac{1}{2}\)[/tex] the length of the segment ensures that the two arcs drawn from the endpoints of the segment will intersect at two unique points, which are necessary for constructing the perpendicular bisector accurately.
