Coordinate Geometry
Coordinate geometry is a branch of mathematics that combines algebra and geometry. It involves studying geometric figures using coordinates and equations. By representing points, lines, and shapes on a coordinate plane, it allows for the application of algebraic techniques to solve geometric problems.
7 Key excerpts on "Coordinate Geometry"
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...13 Analytical Geometry In classical mathematics, analytical geometry (also known as Coordinate Geometry) consists of the study of geometry using a coordinate system as tool – according to which every point is represented by a set of (numerical) coordinates; this approach is widely followed in physics and engineering to manipulate equations of lines and geometric figures, laid on a plane or developing in space. It was independently invented by René Descartes, a French philosopher and mathematician of the seventeenth century; and Pierre de Fermat, a French lawyer and mathematician, who also lived in the seventeenth century – and was further credited for early developments that eventually led to infinitesimal calculus. Of particular interest for process engineering are straight lines and conical curves, and determination of the length of a curve and the surface area bounded by a plane figure, as well as the volume and the outer surface of a solid of revolution; all these topics will be addressed below to some depth. 13.1 Straight Line Despite constituting a primitive concept in mathematics (along with point and plane), a straight line may be seen as the simplest sequence of adjacent points containing (specifically) two given points, say, A and B – so its defining equation should account for a mere two degrees of freedom. Recalling the simplest algebraic functions available – i.e. of the polynomial type as per Eq. (2.135) or Eq. (2.136), one is faced with an equation reading (13.1) where a 0 and a 1 denote the independent and linear coefficients of said polynomial; this situation is illustrated in Fig. 13.1. If the aforementioned two points are characterized by coordinates A (x 0, y 0) and B (x 1,y 1), then they will belong to the straight line at stake provided that (13.2) and (13.3) are simultaneously satisfied – as obtained from Eq. (13.1) after setting x = x 0 and y = y 0, or x = x 1 and y = y 1, respectively; solution of the set of Eqs...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...You knew about the coordinate plane before your geometry course—you just called it the x–y system, rectangular plane, or Cartesian plane. We place two number lines perpendicular to one another at their zero points and call the intersection the origin of the axis system, as shown in Fig. 4.25. This system divides the plane into four quadrants. Both the x and y coordinates of a point are positive in the first quadrant, the x coordinate is negative whereas the y coordinate is positive in the second quadrant, both coordinates are negative in the third quadrant, and the x coordinate is positive whereas the y coordinate is negative in the fourth quadrant. Because the coordinates are always listed as an ordered pair, with the x coordinate first, every address on the coordinate plane can be clearly provided. The name of the x coordinate is abscissa and the name of the y coordinate is ordinate; although these names aren’t used very often, they provide another way to think of the ordered pair in alphabetical order. Other concepts, such as slope and intercepts, are clarified with the visually orientated geometric approach. In algebra we learn about slope by first counting the rise and the run and then learning the slope formula. We learn about intercepts by looking for the place where a line crosses the y axis or the x axis. Fig. 4.25. Coordinate Geometry can be useful in other areas of study, such as statistics. Did you ever think about a histogram as a geometric figure? You need a coordinate plane with a vertical axis and a horizontal axis to construct a histogram, bar chart, or even a pictogram. You also need a coordinate plane if you want a scatter plot. Transformations and Symmetry We will consider rigid transformations, called isometries, because they preserve size and shape. We will not discuss transmography, although these amazing transformations are the basis for many of the special effects for movies, television, and video games...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...4 Analytic Geometry 1. Rectangular Coordinates The points in a plane may be placed in one-to-one correspondence with pairs of real numbers. A common method is to use perpendicular lines that are horizontal and vertical and intersect at a point called the origin. These two lines constitute the coordinate axes; the horizontal line is the x-axis and the vertical line is the y-axis. The positive direction of the x-axis is to the right whereas the positive direction of the y-axis is up. If P is a point in the plane one may draw lines through it that are perpendicular to the x- and y-axes (such as the broken lines of Figure 4.1). The lines intersect the x-axis at a point with coordinate x 1 and the y-axis at a point with coordinate y 1. We call x 1 the x-coordinate or abscissa and y 1 is termed the y-coordinate or ordinate of the point P. Thus, point P is associated with the pair of real numbers (x 1, y 1) and is denoted P (x 1, y 1). The coordinate axes divide the plane into quadrants I, II, III, and IV. FIGURE 4.1. Rectangular coordinates. 2. Distance between Two Points; Slope The distance d between the two points P 1 (x 1, y 1) and P 2 (x 2, y 2) is d = (x 2 − x 1) 2 + (y 2 − y 1) 2 In the special case when P 1 and P 2 are both on one of the coordinate axes, for instance, the x-axis, d = (x 2 − x 1) 2 = | x 2 − x 1 |, or on the. y-axis, d = (y 2 − y 1) 2 = | y 2 − y 1 |. The midpoint of the line segment P 1 P 2 is (x 1 + x 2 2, y 1 + y 2 2). The slope of the line segment P 1 P 2, provided it is not vertical, is denoted by m and is given by m = y 2 − y 1 x 2 − x 1. The slope is related to the angle of inclination α (Figure 4.2) by m = tan α Two lines (or line segments) with...
eBook - ePub- Mark Zegarelli(Author)
- 2011(Publication Date)
- For Dummies(Publisher)
...Coordinates are identified with a pair of axes — the x-axis and the y-axis. The two axes are essentially a pair of number lines that cross at (0, 0), a point that’s also called the origin. Figure 8-1 shows the basic xy- graph with a few points plotted. Each point is identified by a pair of coordinates that includes both an x- value and a y -value, respectively. When plotting a point, start at the origin, find the x- value on the x- axis as you would with a number line, and then count either up (positive) or down (negative) to find the y- value. Figure 8-1: The xy- graph is the basis for Coordinate Geometry. Graphing Linear Functions A linear function is simply a line on the xy- graph. As in plane geometry (which I discuss in Chapter 10), a line is uniquely defined by two points and extends infinitely in both directions. Graphically, a linear function is a straight line extending infinitely in both directions, as shown in Figure 8-2. Figure 8-2: A linear function is a straight line. Linear functions are common on the ACT, so understanding them well is important. In this section, I give you the basic skills you need to answer the most common types of ACT questions that involve lines on a graph — whether or not the question actually includes a graph. Flip to Chapter 9 for practice answering questions that use these skills. Lining up some line segment skills Two useful formulas for answering ACT questions on Coordinate Geometry are the midpoint formula and the distance formula. Both of these formulas provide information about a line segment on a graph between any two points and. In this section, I show you how to use both of these formulas. Finding coordinates with the midpoint formula The midpoint formula allows you to find the coordinates of the midpoint of a line segment between any two points and...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The point of intersection of the axes is called the origin and has coordinates (0, 0). Coordinates An ordered pair of numbers that identifies the location of a point on a coordinate plane, written as (x, y). Origin The point on the coordinate plane where the x - and y -axes intersect; has coordinates (0, 0). Quadrant One of four sections of a coordinate grid separated by horizontal and vertical axes; they are numbered I, II, III, and IV, counterclockwise from the upper right. x-axis The horizontal axis; the line whose equation is y = 0. x-intercept The point where a graph of an equation crosses the x -axis when (y = 0). y-axis The vertical axis; the line whose equation is x = 0. y-intercept The point where a graph of an equation crosses the y -axis when (x = 0). 12.1 What Is Coordinate Geometry? The x -axis runs horizontally, which is left and right. The y -axis is perpendicular to the x -axis and runs vertically, which is up and down. The axes divide the coordinate plane into four equal regions called quadrants. They are numbered with Roman numerals by starting at (+, +) the top-right quadrant and moving in a counterclockwise direction. The points on the plane are found by using ordered pairs of (x, y). The x always comes before the y, just like in the alphabet. Remember, the intersection of two lines is called a point. You can use lines or your finger if it helps. Think, “Find the elevator,” and then go up or down. The point (2, 3) is located by moving from the origin 2 to the right and 3 up. An ordered pair gives direction and distance. The direction is from the sign and the distance is the number. EXAMPLE 12.1 1) Graph and label each of the following points on the coordinate plane...
eBook - ePubUnderstanding Mathematics for Young Children
A Guide for Teachers of Children 3-7
- Derek Haylock, Anne D Cockburn(Authors)
- 2017(Publication Date)
- SAGE Publications Ltd(Publisher)
...9 Understanding Geometry A Mathematical Experience I was watching two boys in the nursery class chasing each other round the playground on tricycles. At high speed they were weaving their way skilfully around all the various obstacles lying around the playground, judging which gaps were large enough to get through, staying within what they understood to be the boundaries for the game and simultaneously relating their route and position to those of the other boy. I asked myself, ‘Were they experiencing mathematics?’ Were they? How important is this kind of informal and intuitive experience of space and shape? In this Chapter In this chapter we endorse the validity and significance of this kind of experience for young children, as providing a foundation for the later development of geometric thinking. We explain how number work and geometric thinking are linked through the two fundamental processes of transformation and equivalence that are at the heart of thinking mathematically. We then provide an analysis of what children will learn about the geometry of space and shape using these two key concepts: looking at all the different ways in which shapes can be transformed, and all the ways in which shapes can be recognized as being in some sense the same, or equivalent. Number and Shape: Two Branches of Mathematics In our view, the two boys on their tricycles described above were undoubtedly engaging in mathematics. Geometry is about describing position and movement in space and recognizing the properties of two- and three-dimensional shapes. Life in a well-equipped nursery is full of such crucial experience of space and shape: building models; playing with construction materials; packing away the toys; putting things in the right place where they fit the available spaces on shelves or in boxes; creating patterns with shapes; rearranging the furniture; moving some objects by pushing and others by rolling; and so on...
eBook - ePubThe Learning and Teaching of Geometry in Secondary Schools
A Modeling Perspective
- Pat Herbst, Taro Fujita, Stefan Halverscheid, Michael Weiss(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...p.48 2 GEOMETRIC FIGURES AND THEIR REPRESENTATIONS 2.1. Introduction The notion of geometric figure, while defined loosely as well as restrictedly by Euclid (“14. A figure is that which is contained by any boundary or boundaries”, Euclid, 1956, p. 153), is central to Euclid’s Elements : Postulates enable possible figure constructions, and propositions assert properties of those figures or demonstrate that other figures can be constructed. Later expositions of geometric knowledge have varied in the extent to which they center on the notion of figure, with modern treatises either making figure a centerpiece as an application of set theory (“by figure we mean a set of points”, Moise, 1974, p. 37) or making figure a derived notion within a more general consideration of geometry as the study of transformations of space onto itself (see for example Guggenheimer, 1967; see also Jones, 2002; Usiskin, 1974). The notion of figure has also played a crucial role in scholarship on the teaching and learning of geometry (e.g., Duval, 1995; Fischbein, 1993). This chapter considers how scholarship from various disciplines has influenced our community’s thinking about geometric figures. It brings perspectives from mathematics and mathematicians, from the history and philosophy of mathematics, from cognitive science and semiotics, from technology and from mathematics education proper. Influenced by those perspectives we elaborate on the role of the geometric figure in geometry teaching and learning. We articulate some conceptions of figure related to its various representations in the context of making a curricular proposal on which to found research and development: To conceptualize the study of geometry in secondary schools as a process of coming to know figures as mathematical models of the experiential world...
