Your friend constructs 𝑋𝑌 so that it is perpendicular to and contains the midpoint of 𝐴𝐵. He claims that 𝐴𝐵 is the perpendicular bisector of 𝑋𝑌. What is his error?
Heck, just draw the line AB. Then pick a point X close to the line and Y far away from the line.
While XY is the pb of AB, AB is not necessarily the pb of XY.
The error in your friend's claim is that constructing XY perpendicular to and containing the midpoint of AB does not automatically make AB the perpendicular bisector of XY.
To explain this, let's first clarify the definition of a perpendicular bisector. A perpendicular bisector is a line or line segment that cuts another line segment into two equal parts while forming right angles with it. So, if AB is the perpendicular bisector of XY, it means that XY is divided into two equal parts, and the angles formed between AB and XY are right angles.
In your friend's construction, XY is perpendicular to AB, which means the angle between XY and AB is 90 degrees. However, this construction does not guarantee that XY is divided into two equal parts. The midpoint of AB could be anywhere along the line XY, and the length of XY between the midpoint and any of its endpoints can be different.
In summary, your friend's error is assuming that constructing XY perpendicular to and containing the midpoint of AB automatically makes AB the perpendicular bisector of XY.
To understand your friend's error, we first need to review what it means for a line to be the perpendicular bisector of another line.
When a line is the perpendicular bisector of another line, it means that it intersects the line at a right angle and also passes through its midpoint.
In this case, your friend correctly constructed line XY perpendicular to line AB. Additionally, they ensured that line XY passed through the midpoint of AB.
However, we cannot conclude that AB is the perpendicular bisector of XY because we need to check if AB intersects XY at a right angle.
To determine whether AB is perpendicular to XY, we need more information about the relationship between these lines. Without any additional information, we cannot confirm or refute your friend's claim.
Therefore, your friend's error lies in making a conclusion without verifying if AB intersects XY at a right angle.