Mathematics
Plane Geometry
Plane geometry is a branch of mathematics that focuses on the properties and relationships of two-dimensional shapes, such as points, lines, angles, and polygons, within a flat surface or plane. It involves the study of concepts like distance, area, perimeter, and symmetry, and is fundamental to understanding spatial relationships and solving geometric problems.
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6 Key excerpts on "Plane Geometry"
eBook - PDFDr. Math Introduces Geometry
Learning Geometry is Easy! Just ask Dr. Math!
- (Author)
- 2004(Publication Date)
- Jossey-Bass(Publisher)
Maybe you’re wondering, then how do we ever build things or make machines that work if we can’t measure things precisely? The answer is that we can usually find a way to measure precisely enough. If my ruler says a piece of paper is 6 inches long and I fold it in half, I know the result will be about 3 inches. A tape measure will tell a good carpenter enough to make a porch that looks square, even, and level, without the carpenter’s knowing its measurements to an accurate hundredth of an inch. But what if perfect forms existed that we could measure pre- cisely? They do in our minds. These are what we study in geometry. Geometry has applications in the physical world, and its principles have made it possible for us to build amazing things from our imper- fect materials and measurements. This book will introduce you to the definitions and properties of 1 Introduction A B C D two-dimensional objects, including squares, rectangles, and cir- cles. You’ll learn how to work with them and how changing one of their dimensions changes other dimensions. You’ll also learn about three-dimensional objects: what properties they have in common with two-dimensional forms and what sets them apart. Finally, we’ll talk about patterns on surfaces, specifically symmetry and tessel- lations in two dimensions. Before you know it, you’ll be seeing perfect geometry all around you. Dr. Math welcomes you to the world and language of geometry! 2 Introduction T wo-dimensional geometry, coordinate Plane Geometry, Cartesian geometry, and planar (pronounced PLANE-er) geometry refer to the same thing: the study of geometric forms in the coordinate plane. Do you remember the coordinate plane? It’s a grid system in which two numbers tell you the location of a point—the first, x, tells you how far left or right to go from the origin (the center point), and the second number, y, tells you how far up or down to go. The y-axis is vertical and the x-axis is horizontal (like the horizon).
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
F u n d a m e n t a l s o f P l a n e G e o m e t r y S E C T I O N I V Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 518 OBJECTIVES After studying this unit you should be able to ■ identify axioms and postulates that apply to geometric statements. ■ write geometric statements in symbol form. ■ illustrate geometric statements. Geometry is the branch of mathematics in which the properties of points, lines, surfaces, and solids are studied. Many geometric principles were first recognized by the Babylonians and Egyptians more than 5000 years ago. The Egyptians used geometry for land survey-ing. This is the earliest known use of geometry. Throughout the centuries, geometry has been used in many ways that have greatly influenced modern living. Much of the environment has been affected by the use of geometric principles. Practically everything in modern living depends on geometry. Geometric applications are used in building houses, apartments, offices, and shops. Roads, bridges, and air-ports could not be constructed without the use of geometry. Automobiles, airplanes, and ships could not be designed and produced without the application of geometric principles. The manufacture of clothes and the processing and distribution of food depend on geometric applications. Many occupations require a knowledge of geometry and the ability to apply this knowledge to practical on-the-job uses. Carpentry, plumbing, machining, draft-ing, and auto body repair are but a few of the occupations in which geometry is used regularly.
eBook - PDF- Maria Catherine Borres(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
BASIC CONCEPTS OF ANALYTIC GEOMETRY CHAPTER 1 CONTENTS 1.1. Introduction ........................................................................................ 2 1.2. Distance Between Two Points ............................................................. 4 1.3. Equations of A Straight Line ................................................................ 7 1.4. Direction Angles of A Vector ............................................................. 11 1.5. Angle Between Two Straight Lines ..................................................... 12 1.6. Distance of A Point From A Line ....................................................... 14 1.7. Equations For Planes ......................................................................... 16 1.8. Plane Through Three Given Non-Collinear Points .............................. 18 1.9. Angle Between Two Planes ............................................................... 26 1.10. Straight Lines As Intersection of Two Planes .................................... 27 1.11. A Straight Line And A Plane ............................................................ 29 Principles of Applied Mathematics 2 1.1. INTRODUCTION In analytic geometry of two dimensions the underlying space is 2 × = R R R and each point of 2 R is an ordered pair ( ) , x y of real numbers. In ana-lytic geometry of three dimensions we consider the space × × R R R or sim-ply 3 . R Any element of 3 R is an ordered triple ( ) , , , x y z where , , x y z are real numbers. As in the case of element of 2 , R two ordered triples ( ) ( ) , , , , x y z and a b c belonging to 3 R are equal if, and only if, their cor-responding components are equal i.e., if, and only if , . = = = x a y b and z c A coordinate system in 3 R is defined as under. Definition: We select a fixed point O for the origin. The x and y axes are drawn mutually perpendicular through O as in the case of plane analytic geometry.
eBook - PDFScalar, Vector, and Matrix Mathematics
Theory, Facts, and Formulas - Revised and Expanded Edition
- Dennis S. Bernstein(Author)
- 2018(Publication Date)
- Princeton University Press(Publisher)
Chapter Five Geometry 5.1 Facts on Angles, Lines, and Planes Fact 5.1.1. Let X △ = { x 1 , . . . , x n } be a set of points in R 2 , and assume that no three points in X lie in a single line. Furthermore, let L △ = { L 1 , . . . , L n ( n -1) / 2 } be the set of lines passing through all pairs of points in X , and assume that no pair of lines in L is parallel and no three of the lines in L intersect at a point that is not in X . Let P denote the set of polygons whose boundaries are subsets of ∪ n ( n -1) / 2 i = 1 L i and whose interiors are disjoint from ∪ n ( n -1) / 2 i = 1 L i . Then, the lines in L intersect at exactly 1 8 n ( n -1)( n -2)( n -3) points that are not in X , and the number of polygons in P is 1 8 ( n -1)( n 3 -5 n 2 + 18 n -8) , of which n ( n -1) are not bounded. Source: [771, p. 72]. Fact 5.1.2. Let L △ = { L 1 , . . . , L n } be a set of lines in R 2 , assume that no pair of lines in L is parallel, and assume that the intersection of each triple of lines in L is empty. Let P denote the set of polygons whose boundaries are subsets of ∪ n i = 1 L i and whose interiors are disjoint from ∪ n i = 1 L i . Then, the number of polygons in P is 1 2 ( n 2 + n + 2) , of which 1 2 ( n -1)( n -2) are bounded. Source: [771, p. 72]. Fact 5.1.3. Let P △ = { P 1 , . . . , P n } be a set of planes in R 3 , assume that no pair of planes in P is parallel, and assume that the intersection of each triple of planes in P is a single point. Let H denote the set of polyhedra in R 3 whose boundaries are subsets of ∪ n i = 1 P i and whose interiors are disjoint from ∪ n i = 1 P i . Then, the number of polyhedra in H is ∑ 3 i = 0 ( n i ) = 1 6 ( n 3 + 5 n + 6) , of which ( n -1 3 ) = 1 3 ( n -1)( n -2)( n -3) are bounded. Source: [771, p. 72]. Remark: Extensions to hyperplanes in R n are discussed in [771, p. 72]. Fact 5.1.4. The points x , y , z ∈ R 2 lie on one line if and only if det [ x y z 1 1 1 ] = 0 .
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
When you’ll use your knowledge of proofs Will you ever use your knowledge of geometry proofs? I’ll give you a politically correct answer and a politically incorrect one. Take your pick. First, the politically correct answer (which is also actually correct). Granted, it’s extremely unlikely that you’ll ever have occasion to do a single geometry proof outside of a high school maths course. However, doing geometry proofs teaches you important lessons that you can apply to nonmathematical arguments. Proofs teach you 6 Not to assume things are true just because they seem true. 6 To carefully explain each step in an argument even if you think it should be obvious to everyone. 6 To search for holes in your arguments. 6 Not to jump to conclusions. In general, proofs teach you to be disciplined and rigorous in your thinking and in communicating your thoughts. If you don’t buy that PC stuff, I’m sure you’ll get this politically incorrect answer: Okay, so you’re never going to use geometry proofs, but you want to get a decent grade in maths, right? So you might as well pay attention in class (what else is there to do, anyway?), do your homework, and use the hints, tips and strategies I give you in this book. They’ll make your life much easier. Promise. Getting Down with Definitions The study of geometry begins with the definitions of the five simplest geometric objects: Point, line, segment, ray and angle. And I throw in two extra definitions for you (plane and 3‐D space) for no extra charge. 340 Part IV: Applying Algebra and Understanding Geometry 6 Point: A point is like a dot except that it has no size at all. A point is zero‐dimensional, with no height, length or width, but you draw it as a dot anyway. You name a point with a single uppercase letter, as with points A, D and T in Figure 16‐2. 6 Line: A line is like a thin, straight wire (although really it’s infinitely thin — or better yet, it has no width at all).
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
chapter 2 Combining Algebra and Geometry Analytic geometry, which combines algebra and geometry, provides a tremen- René Descartes, who invented the coordi- nate system we use to graph equations, explaining his work to Queen Christina of Sweden (from an 18 th - century painting by Dumesnil). dously powerful tool for visualizing equations with two variables. We begin this chapter with a description of the coordinate plane, including a discussion of distance and length. Then we turn our attention to lines and their slopes, which are simple concepts that have immense importance. We then investigate quadratic expressions. We will see how to complete the square and solve quadratic equations. Quadratic expressions will also lead us to the conic sections (ellipses, parabolas, and hyperbolas). We conclude the chapter by learning methods for computing the area of triangles, trapezoids, circles, and ellipses. 41 42 chapter 2 Combining Algebra and Geometry 2.1 The Coordinate Plane learning objectives By the end of this section you should be able to locate points in the coordinate plane; graph equations with two variables in the coordinate plane, possibly using technology; compute the distance between two points; compute the circumference of a circle. Coordinates Recall how the real line is constructed: We start with a horizontal line, pick a point on it that we label 0, pick a point to the right of 0 that we label 1, and then we label other points using the scale determined by 0 and 1 (see Section 1.1 to review the construction of the real line). The coordinate plane is constructed in a similar fashion, but using a horizontal and a vertical line rather than just a horizontal line. 3 2 1 1 2 3 3 2 1 1 2 3 The coordinate plane, with a dot at the origin. The coordinate plane • The coordinate plane is constructed by starting with a horizon- tal line and a vertical line in a plane.
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