Answer:
The conjecture that the set of points equidistant from A and B is the perpendicular bisector of segment AB is indeed true, and it's essentially what's being demonstrated when you fold the paper to align A and B.
Explanation:
Short Answer:
When you fold a sheet of paper, aligning points A and B, you create a crease that serves as the fold line. This crease becomes the perpendicular bisector of segment AB. A perpendicular bisector is a line that intersects a segment at a right angle and divides it into two equal parts. Folding the paper aligns A and B, naturally making the crease perpendicular to segment AB. Additionally, since you're matching A and B exactly, the crease runs through the midpoint of AB. This ensures that the crease bisects AB into two equal segments. Ultimately, folding the paper in this manner guarantees that the crease is both perpendicular to AB and passes through its midpoint, thus satisfying the criteria of a perpendicular bisector for segment AB.
In depth:
When you fold a sheet of paper so that points A and B match up with each other, you are essentially creating a crease that acts as the fold line. The question asks you to explain why this crease is the perpendicular bisector of segment AB.
Perpendicular: The perpendicular bisector of a line segment is a line that is perpendicular (at a right angle) to the line segment. When you fold the paper along the crease, the fold line will naturally be perpendicular to the line segment AB because the process of folding involves aligning points A and B.
Bisector: The bisector part refers to dividing the line segment AB into two equal parts. When you fold the paper, the crease goes through the midpoint of segment AB. This is because you're aligning point A with point B perfectly. Since the crease goes through the midpoint of AB, it divides AB into two equal segments.
So, by folding the paper so that points A and B match up, you're ensuring that the fold line (crease) is both perpendicular to AB and passes through its midpoint, making it the perpendicular bisector of segment AB.