Question

Select all statements that are always true for line segment AB as the perpendicular bisector of line segment XY with point P as the midpoint of line segment XY:

a) AY = XB
b) XP = YP
c) AP = XP
d) AB = XY

Answer 1

In the context of a perpendicular bisector AB of segment XY with midpoint P, statements a) 'AY = XB' and b) 'XP = YP' are always true because they relate to the properties of a perpendicular bisector and midpoint, respectively. Statements c) and d) are not always true, as they imply relationships between line segments that are not defined by the perpendicular bisector.

Step-by-step explanation:

If line segment AB is the perpendicular bisector of line segment XY, with point P as the midpoint of XY, we can analyze the given statements:

• a) AY = XB: This is true because AB, being the perpendicular bisector, means that it creates two congruent triangles, ΔAPY and ΔBPX, where AY = XB due to the congruency of the corresponding sides of the triangles.
• b) XP = YP: This is also true because P is the midpoint of XY, meaning XP = PY by definition of a midpoint.
• c) AP = XP: This is not necessarily true. AP is a segment of triangle ΔAPY, and XP is a segment from the midpoint to one end of XY. As ΔAPY need not be isosceles, AP does not necessarily equal XP.
• d) AB = XY: This statement is not always true. A perpendicular bisector divides the segment into two equal parts, but does not give any information about its own length relative to XY.

In conclusion, statements a) and b) are always true for a perpendicular bisector with the conditions given, while statements c) and d) are not guaranteed to be true.