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Geometry

This section reviews the coordinate plane, axis division, ordered pairs, & their relation to algebraic equations. Knowledge of algebra is essential for geometric analysis.

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Geometry

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  1. Geometry Integration Algebra: The Coordinate Plane Section 1-1

  2. This section is a review of the coordinate plane. It deals with definitions and concepts that you have already studied in Algebra I. …\GeoSec01_01.ptt

  3. The coordinate plane is a visual representation of a binary relationship. The coordinate plane we will use in this course is defied by two Real number lines that are perpendicular to each other. Each of the two number lines are called an axis. The verticalaxis is called the y-axis and the horizontal axis is called the x-axis. The intersection of the two real number lines is called the origin and is labeled as zero on both axis. From the origin, the y-axisincreasesin value as you move up the axis and decreases in value as you proceed down the axis. The x-axisincreasesin valuefrom left to right and decreases in value as you move from right to left. …\GeoSec01_01.ptt

  4. y 5 4 3 2 1 0 -1 -2 -3 -4 -5 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 The axes divide the coordinate plane into four quadrants. Quadrant II Quadrant I Quadrant III Quadrant IV …\GeoSec01_01.ptt

  5. A relation is developed when we associate a given x-axis value with a given y-axis value. We call this relation an ordered pair and it is symbolized by two numbers enclosed in parenthesis, (x, y). The x-axis part of the ordered pair is called the x-coordinate and the y-axis part is called the y-coordinate. Together they are called the coordinates of the ordered pair. The relation is called an ordered pair since the order of the two numbers makes a difference. As shown, the x-axis value is the first number and the y-axis value is the second number in the ordered pair. Thus, the topic of this lesson, the coordinate plane. …\GeoSec01_01.ptt

  6. For example, the x-axis could be your daily grade and the y-axis could be the price of tea in China. This relationship may be logically flawed, but the setup of the coordinate plane to plot this relationship is acceptable. In this course we will deal mostly with a coordinate plane composed of two real number axes. However, as you have learned or will learn in other courses, the coordinate plane can be used to represent any binary relationship. …\GeoSec01_01.ptt

  7. Throughout this course you will be learning geometry, but algebra skills are absolutely essential to learning the material in this course. Algebra is used to analyze the geometry and to quantify (i.e., a numerical sense or feel) the geometry. In turn, the geometry gives the numerical aspects of algebra a visual perspective. With this thought in mind, our discussion turns to a quick overview of how algebraic relations can be expressed. …\GeoSec01_01.ptt

  8. y table of values graph 5 4 3 2 1 0 -1 -2 -3 -4 -5 x • E -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 • D x -2 -1 0 2 3 y -5 -3 -1 3 5 (x, y) A(-2, -5) B(-1,-3) C(0, -1) D(2, 3) E(3, 5) • C • B • A equation y = 2x - 1 …\GeoSec01_01.ptt

  9. y 5 4 3 2 1 0 -1 -2 -3 -4 -5 x • E -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 • D • C • B • A Suppose I want to determine if the ordered pair (8,17) belong to the solution set. I would substitute the ordered pair x-coordinate, and y-coordinate into the equation. y = 2x – 1 becomes 17 = 2  8 -1 17  15 Suppose I want to determine if the ordered pair (7,13) belong to the solution set. I would substitute the ordered pair x-coordinate, and y-coordinate into the equation. y = 2x – 1 becomes 13 = 2  7 -1 13 = 13 An important part of this graph is the points A, B, C, D, and Esatisfy the equation y = 2x - 1. The graph of the equation is the line that passes through all the points A, B, C, D, and E. Since they are all on the same line, they are by definition collinear. To determine if a coordinatesatisfies the equation, substitute the coordinate values for x and y in the equation. If the equation balances, then the coordinate belongs to the solutionset of the equation. …\GeoSec01_01.ptt

  10. In this section we reviewed the coordinate plane and the terminology associated with the coordinate plane. Terminology is essential to the study of geometry. You then learned that algebraic relations, which will be used throughout this course, can be expressed in three ways; 1) in an equation, 2) in a table, and 3) with a graph. You must be able to talk and understand the language of geometry in order to study and learn geometry. The terminology learned here will be used throughout this course. We also covered the definition of collinear and discussed the terms solution sets and balanced equations. …\GeoSec01_01.ptt

  11. END OF LINE …\GeoSec01_01.ptt

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