You have worked with lines and line segments in past courses.
Let's refresh some basic facts we know to be true about segments.


definition
A segment (or line segment) is a part of a line that is bounded by two distinct end points. It contains every point on the line between its end points.
segment
segmentnotation

Postulate:  A line segment is the shortest distacnce between two distinct points.

Ruler Postulate

Ruler Postulate: The points on a line can be put into a one-to-one correspondence (paired) with the real numbers. The distance between any two points is represented by the absolute value of the difference between the numbers. [Keep in mind that distances are always positive.]

ruler

The distance between D and F is | 0 - 2 | = 2.

The distance between C and E is | -2 - 1 | = 3.

The distance between A and C is | -3 - (-2) | = 1.

 

Segment Addition Postulate

Statement: If B lies on the segment from A to C, then AB + BC = AC.
Also the converse: If AB + BC = AC, then B lies on the segment from A to C.

You may see examples where the concept of this postulate is referred to as "whole quantity", "the whole is equal to the sum of its parts" or "betweenness of points". Be sure to use the statement that your teacher wishes to follow.

In this postulate, points A, B and C are collinear, meaning they all lie on the same line.

 

Midpoint of a Segment

definition
The midpoint of a segment is a unique point on the segment forming two congruent segments.

Example:

midptexample

midptexG

midANS

 

Bisector of a Segment

definition
The bisector of a segment is a line, ray, or segment which cuts the given segment into two congruent segments.
The point where the bisector crosses the segment is the midpoint of the segment.
bisector3
There are an infinite number of possible bisectors of a segment passing through the midpoint.

If the bisector is also perpendicular to the segment, it is referred to as the perpendicular bisector of the segment.

While there are an infinite number of possible bisectors
of a segment, there is only one perpendicular bisector.

perpbis


Theorem: The points on a perpendicular bisector are equidistant from the endpoints of the line segment.

In the diagram, line m is the perpendicular bisector of the segment from A to B.

Every point on line m is the same distance from A, as it is from B.

Distances:   AC = CB       AD = DB       AE = EB



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