Question

a sheet of paper with points a and b is folded so that a and b match up with each other.
a • • b
explain why the crease in the sheet of paper is the perpendicular bisector of segment ab. (assume the conjecture that the set of points equidistant from a and b is the perpendicular bisector of segment ab is true.)
select all the true statements.
when the paper is folded so that a lies directly on top of b, every point on the crease is the same distance from a and b. since the set of all such points is the perpendicular bisector of ab, the crease must be that perpendicular bisector.
folding the paper so that a matches up with b means that the crease is the line of symmetry between them. a line of symmetry places points equally distant from a and b, which is exactly the definition of the perpendicular bisector of ab
the crease is the perpendicular bisector of ab because it passes through both points a and b. since the fold lines up the two points, the crease must go directly through them.
if c is a point on the crease of the folded paper, a segment can be drawn that simultaneously connects c to a and b while the paper is still folded. it is the same segment so this means that the distance from c to a is the same as the distance from c to b. since every point on the crease is the equidistant to a and b, the crease is the perpendicular bisector of segment ab.

Explanation:

1. First statement: When folded, points on crease are equidistant from A and B. Given the conjecture, this set is the perpendicular bisector, so this is true.
2. Second statement: Crease is line of symmetry, placing points equidistant from A and B, matching the perpendicular bisector definition, so true.
3. Third statement: The crease does not pass through A and B (it bisects the segment between them), so this is false.
4. Fourth statement: For any point C on crease, CA = CB (since folded, they coincide), so all crease points are equidistant from A and B, hence perpendicular bisector, true.

Answer:

• When the paper is folded so that A lies directly on top of B, every point on the crease is the same distance from A and B. Since the set of all such points is the perpendicular bisector of AB, the crease must be that perpendicular bisector.
• Folding the paper so that A matches up with B means that the crease is the line of symmetry between them. A line of symmetry places points equally distant from A and B, which is exactly the definition of the perpendicular bisector of AB
• If C is a point on the crease of the folded paper, a segment can be drawn that simultaneously connects C to A and B while the paper is still folded. It is the same segment so this means that the distance from C to A is the same as the distance from C to B. Since every point on the crease is equidistant to A and B, the crease is the perpendicular bisector of segment AB.