True or False?

1.) A segment has exactly one perpendicular bisector.

2.) A line has exactly one perpendicular bisector.

3.)Given a line and a point not on the line, there is exactly one line perpendicular to the original line that passes through the given point.

joe mama false

1. A line segment has ... true

2. false, the end-points of the line is not defined.
3. true (even if the point is on the line)

1.) True. Just make sure the segment behaves and doesn't start dividing itself into multiple perpendicular bisectors. Otherwise, it's off to geometry detention!

2.) False. Lines are a bit more rebellious and have infinite perpendicular bisectors. They just can't make up their mind about which one to choose!

3.) True. As long as the given point promises not to lead the line astray, there will always be one line that plays by the rules and goes perpendicular through it. No funny business here!

1.) True.

To determine if a segment has a perpendicular bisector, we need to consider the definition of a perpendicular bisector. A perpendicular bisector is a line that divides a segment into two equal parts, while also being perpendicular to the segment.

To find the perpendicular bisector of a segment, we can follow these steps:
1. Identify the mid-point of the segment by using the midpoint formula, which is (x1 + x2) / 2, (y1 + y2) / 2 for a segment with endpoints (x1, y1) and (x2, y2).
2. Calculate the slope of the segment using the formula (y2 - y1) / (x2 - x1).
3. Calculate the negative reciprocal of the slope obtained in step 2 to find the perpendicular slope. This is done by flipping the fraction and changing the sign.
4. Apply the point-slope form of a line, using the mid-point obtained in step 1 and the perpendicular slope calculated in step 3 to define the equation of the perpendicular bisector.

Since there is only one mid-point for a given segment, and only one equation can be formed using the point-slope form, it follows that a segment has exactly one perpendicular bisector.

2.) False.

As opposed to a segment, a line does not have endpoints, and therefore, it does not have a mid-point. Without a mid-point, the point-slope form cannot be used to find a unique equation for a perpendicular bisector.

Hence, a line does not have exactly one perpendicular bisector.

3.) True.

Given a line and a point not on the line, there is exactly one line perpendicular to the original line that passes through the given point. To understand why, we can consider the definition of perpendicular lines.

Two lines are perpendicular if they intersect at a 90-degree angle. The slope of a line is directly related to its angle of inclination. Perpendicular lines have slopes that are negative reciprocals of each other.

Suppose we have a line and a point not on the line. We can draw an infinite number of lines passing through the given point, but only one of them will be perpendicular to the original line. This is because the slopes of any two perpendicular lines will always multiply to -1.

By finding the slope of the given line and then calculating the negative reciprocal, we can determine the unique slope of the line perpendicular to the original line. Finally, using the point-slope form with the given point and the slope, we can find the equation of the perpendicular line.

A segment has exactly one perpendicular bisector

joe mama true