Technical Mathematics, Sixth Edition

22.1. The Straight Line

Let us begin our study of analytic geometry with the familiar straight line. We already graphed the straight line in an earlier chapter, and here we build upon what we did there. There is no need to point out the usefulness of the straight line in technology, so whatever new tools we can find to work with the straight line should be welcome. We will first learn how to calculate the length of a line segment. Recall that earlier we defined a line segment as the portion of a straight line lying between two endpoints.

22.1.1. Length of a Line Segment Parallel to a Coordinate Axis

The length of a line segment is the distance between its endpoints. By length we mean the magnitude of that length; thus it is always positive.

▪ Exploration:

Try this. Without reading further, find the lengths of the three lines in Fig. 22-2.

It should be no surprise that to find the length of a line segment lying on or parallel to the x axis, we simply subtract the x value at one endpoint from that at the other endpoint, and take the absolute value of the result. The procedure is similar for lines on or parallel to the y axis.

Figure 22.2. FIGURE 22-2

Example 1: The magnitudes of the lengths of the line segments in Fig. 22-2 are: AB = |5 − 2| = 3 CD = |5 − (−2)| = 7 RS = |−1 − (−5)| = 4 Note that it doesn't matter if we reverse the order of the endpoints. We get the same result. ...