Coordinate Geometry

3 Key excerpts on "Coordinate Geometry"

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  eBook - ePub

  A First Course in Geometry

  • Edward T Walsh(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  As far as the propositions of mathematics refer to reality they are not certain; and as far as they are certain, they do not refer to reality.

  ALBERT EINSTEIN (1879–1955)

  CHAPTER TEN

  Coordinate Geometry

  10.1  INTRODUCTION

       By the early seventeenth century, the vast expansion of science and commerce had raised mathematical problems that, in their geometric formulation, seemed nearly impossible to solve: problems designing lenses for telescopes, problems concerning the motion of projectiles, and the like. Then two Frenchmen, René Descartes and Pierre de Fermat (1601–1665), working independently of each other, realized the potentialities inherent in the algebraic representation of geometric curves. They developed new systematic methods for dealing with curves, and showed how helpful it is to be able to attack geometric problems algebraically and vice versa.

       No doubt you recall how, in Chapter 2 , the concept of the number line was introduced and geometric methods of solving certain complicated algebraic problems were developed. In this chapter we will extend the notion of a coordinate system to two dimensions-—to the plane. Such a study is called Coordinate Geometry or analytic geometry.

  10.2  COORDINATE SYSTEMS AND DISTANCE

       In Section 2.4 we established the existence of a coordinate system, a particular type of one-to-one correspondence between the real numbers and the points on a line. Recall the Ruler Placement Postulate:

       POSTULATE 10 Let be a line with P ∈ , Q ∈ , and P ≠ Q. Then there exists a coordinate system for such that c(P) = 0 and c(Q) is positive.

       An example of such a coordinate system is depicted in the accompanying figure.

       We will now use a method suggested by the work of Descartes to establish a coordinate system for a plane.

       First we introduce a line with a given coordinate system. We call this line the x-axis, and it is generally drawn horizontally. Next we introduce a second line , perpendicular to the first line at the point with coordinate 0. This second line is called the y-axis. We establish a coordinate system on the y-axis, assigning the coordinate 0 to the point of intersection of the two axes. The positive direction on the x-axis is generally taken to be to the right, while on the vertical y-axis it is upward. It is conventional to adopt the same scale on both axes; that is, the unit of distance is the same. Further, we call the intersection of the two axes the origin, and usually label it with the letter O

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  eBook - ePub

  Geometry For Dummies

  • Mark Ryan(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  Part 7

  Placement, Points, and Pictures: Alternative Geometry Topics

  IN THIS PART … Explore Coordinate Geometry. Discover reflections, translations, and rotations. Tackle locus problems. Passage contains an image Chapter 18

  Coordinate Geometry

  IN THIS CHAPTER Finding a line’s slope and a segment’s midpoint Calculating the distance between two points Doing geometry proofs with algebra Working with equations of lines and circles

  In this chapter, you investigate the same sorts of things you see in previous chapters: perpendicular lines, right triangles, circles, perimeter, area, the diagonals of quadrilaterals, and so on. What’s new about this chapter on Coordinate Geometry is that these familiar geometric objects are placed in the x-y coordinate system and then analyzed with algebra. You use the coordinates of the points of a figure — points like or — to prove or compute something about the figure. You reach the same kind of conclusions as in previous chapters; it’s just the methods that are different.

  The x-y, or Cartesian, coordinate system is named after René Descartes (1596–1650). Descartes is often called the father of Coordinate Geometry, despite the fact that the coordinate system he used had an x- axis but no y- axis. There’s no question, though, that he’s the one who got the ball rolling. So if you like Coordinate Geometry, you know who to thank (and if you don’t, you know who to blame).

  Getting Coordinated with the Coordinate Plane

  I have a feeling that you already know all about how the x-y coordinate system works, but if you need a quick refresher, no worries. Figure 18-1 shows you the lay of the land of the coordinate plane.

  © John Wiley & Sons, Inc.

  FIGURE 18-1: The x-y coordinate system.

  Here’s the lowdown on the coordinate plane you see in Figure 18-1 :

  • The horizontal axis, or x -axis, goes from left to right and works exactly like a regular number line. The vertical axis, or y -axis, goes — ready for a shock? — up and down. The two axes intersect at the origin

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  eBook - ePub

  Fundamental Concepts of Geometry

  • Bruce E. Meserve(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 1 FOUNDATIONS OF GEOMETRY The word “geometry” is derived from the Greek words for “earth measure.” Since the earth was assumed to be flat, early geometers considered measurements of line segments, angles, and other figures on a plane. Gradually, the meaning of “geometry” was extended to include the study of lines and planes in the ordinary space of solids, and the study of spaces based upon systems of coordinates, as in analytic plane geometry, where points are represented by sets of numbers (coordinates) and lines by sets of points whose coordinates satisfy linear equations. During the last century, geometry has been still further extended to include the study of abstract spaces in which points, lines, and planes may be represented in many ways. We shall be primarily concerned with the fundamental concepts of the ordinary high-school geometry — euclidean plane geometry. Our discussion of these concepts is divided into three parts: the study of the foundations of mathematics (Chapter 1), the development of euclidean plane geometry from the assumption of a few fundamental properties of points and lines (Chapters 2 through 6), and a comparison of euclidean plane geometry with some other plane geometries (Chapters 7 through 9). The treatment of the second part —the development of euclidean plane geometry — forms the core of this text and emphasizes the significance of the assumptions underlying euclidean geometry. Together, the three parts of our study enable us to develop an understanding of and appreciation for many fundamental concepts of geometry. 1–1 Logical systems. We shall consider geometries as logical systems. That is, we shall start with certain elements (points, lines,. . .) and relations (two points determine a line,. . .) and try to deduce the properties of the geometry

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