1.3Segments 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

A ray is a point on a line and all points on that line which lie on one side of that point. The ray below is point A and all points on line that lie on the same side of A as B.

How to Name a RayTo name a ray, we first write the name of its endpoint and then follow with the name of a second point on it. Over the two, we place the ray symbol - . Order matters!

If A is the end-point of a ray and B a second point on it, the name of that ray is: __________.

Length Defined

The length (or sometimes measure) of a line segment is the distance from one of its endpoints to the other.

If A and B are the endpoints of a segment, we designate its length as AB or as m. Notice the lack of the segment symbol in the first. The ‘m’ in the first should be read as ‘measure’.

Illustration

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

Hash MarksIf 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 PostulateFirst 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 |rA - rB|, where again rA and rB 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

Once we have a 0-point and a 1-point, we can lay off another 1 cm and get our 2-point, and then another and get our 3-point, etc. Numbers between integers (0.5, , , , etc.) are placed in their appropriate spots.

We now have half of a number line.

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 ImportantThe 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 PostulateLet 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