1.3 Segments and their Measures Today we give our official definitions of segment and ray. We also state the Ruler Postulate. It is the bridge that connects geometry and algrebra. Textbook Section 1-2

Line Segment Defined A line segment is two points on a line and all points that lie between them. We call those two points the end-points of the segment. Illustration:

How to Name a Segment To name a line segment, we write the names of its two endpoints side-by-side and then place a little segment on top. Order is irrelevant. Thus if A and B are the end-points of a line segment, that segment’s name is: __________ or __________.

Ray Defined

How to Name a Ray

Length Defined

Illustration The line segment is below is named ________ or ________. The length of the line segment is ________ or ________.

Hash Marks If we wish a diagram to say that a pair of segments have the same length, we do so with hash marks. The diagram below says that segments AB and CD have the same length. More than one hash mark may be used.

The Computation of Length In our definition of distance, we assumed that for any two points, there’s a unique positive real number that represents the distance between them. This means that for every line segment, there’s a unique positive real that represents its length. Where does that real number come from? The Ruler Postulate gives us our answer.

The Ruler Postulate First I’ll state the Ruler Postulate. After I’ll explain it. We may place the points of a line in correspondence with the real numbers so that: 1.Each point on the line corresponds to a unique real, and each real corresponds to a unique point. 2.The distance between and two points A to B is |r A - r B |, where again r A and r B are the reals assigned to A and B respectively.

0 and 1 How is the correspondence of points of reals established? We choose a unit of measure, perhaps the inch or the centimeter. We choose a 0 point. We choose a direction from the 0 point and mark a ‘1’ one unit from 0.

The Positives

Illustration

The Negatives On the other side of 0, we do just the same except of course that the numbers are now negative. The result is the complete number line. Here’s a piece:

Coordinates and Distance We say that the coordinate of a point on a number line is the real number assigned to it. We define the distance between two points as the absolute value of the difference of their coordinates. Note a consequence: distance is always positive.

Illustration

Why It’s Important The Ruler Postulate turns points into numbers, and that allows us to import the techniques of algebra into geometry. Without the Ruler Postulate, geometry and algebra would have little or nothing to do with one another. With the Ruler Postulate, we have a way to translate between the language of geometry and the language of algebra. That’s a Good Thing.

The Segment Addition Postulate Let us add a second postulate. It’s a natural extension of the Ruler Postulate. The Segment Addition Postulate: if P lies between points A and B, the sum of the lengths of sub-segments AP and BP equals the length of segment AB, i.e. AP + PB = AB.

Generalized We will assume that the Segment Addition Postulate applies no matter the number of points between the endpoints of a line segment. Thus in the diagram below, AB = AP + PQ + QR + RB.

Typical Problem Find the value of x, the length of AP and the length of PB if P lies between A and B, AB = 12, AP = x and PB = 2x - 6.

Midpoint Defined The midpoint of a line segment is that point between its endpoints that splits it up into a pair of segments of equal length. More precisely: if M is the midpoint of segment AB, then M is that point between A and B such that AM = BM. Illustration:

Illustration