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# Points, Lines, and Rays Help (page 2)

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### Rays (Half Lines)

Sometimes, mathematicians talk about the portion of a geometric line that lies “on one side” of a certain point. In Fig. 1-1, imagine the set of points that starts at P , then passes through Q , and extends onward past Q forever. This is known as a ray or half line.

Fig. 1-1 . Two point principle. For two specific points P and Q , line L is unique.

The ray defined by P and Q might include the end point P , in which case it is a closed-ended ray . If the end point is left out, the theoretical object is an open-ended ray . In either case, the ray is said to “begin” at point P ; informally we might say that it is either “tacked down at the end” or “dangling at the end.”

### Midpoint Principle

Suppose there is a line segment connecting two points P and R . Then there is one and only one point Q on the line segment such that PQ = QR , as shown in Fig. 1-2.

Fig. 1-2 . Midpoint principle. Point Q is unique.

## Points, Lines, and Rays Practice Problems

#### PROBLEM 1

Suppose, in Fig. 1-2, we find the midpoint Q 2 between P and Q , then the midpoint Q 3 between P and Q 2 , then the midpoint Q 4 between P and Q 3 , and so on. In mathematical language, we say we keep finding midpoints Q ( n +1) between P and Q n , where n is a positive whole number. How long can this process go on?

Fig. 1-2 . Midpoint principle. Point Q is unique.

#### SOLUTION 1

The process can continue forever. In theoretical geometry, there is no limit to the number of times a line segment can be cut in half. This is because a line segment contains an infinite number of points.

#### PROBLEM 2

Suppose we have a line segment with end points P and Q . What is the difference between the distance PQ and the distance QP ?

#### SOLUTION 2

This is an interesting question. If we consider distance without paying attention to the direction in which it is measured, then PQ = QP . But if direction is important, we define PQ = − QP .

In basic plane geometry, direction is sometimes specified in diagrams in order to get viewers to move their eyes from right to left instead of from left to right, or from bottom to top rather than from top to bottom.

Practice problems of these concepts can be found at: Geometry Basic Rules Practice Test.

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