The Geometry of a Circle
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Analytic Geometry
Guide Study Georgia End-Of-Course Tests Georgia ANALYTIC GEOMETRY TABLE OF CONTENTS INTRODUCTION ...........................................................................................................5 HOW TO USE THE STUDY GUIDE ................................................................................6 OVERVIEW OF THE EOCT .........................................................................................8 PREPARING FOR THE EOCT ......................................................................................9 Study Skills ........................................................................................................9 Time Management .....................................................................................10 Organization ...............................................................................................10 Active Participation ...................................................................................11 Test-Taking Strategies .....................................................................................11 Suggested Strategies to Prepare for the EOCT ..........................................12 Suggested Strategies the Day before the EOCT ........................................13 Suggested Strategies the Morning of the EOCT ........................................13 Top 10 Suggested Strategies during the EOCT .........................................14 TEST CONTENT ........................................................................................................15 -
Chapter 11. Three Dimensional Analytic Geometry and Vectors
Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
Lesson 3: Rectangles Inscribed in Circles
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection . -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Writing the Equation of a Line
Name______________________________ Writing the Equation of a Line When you find the equation of a line it will be because you are trying to draw scientific information from it. In math, you write equations like y = 5x + 2 This equation is useless to us. You will never graph y vs. x. You will be graphing actual data like velocity vs. time. Your equation should therefore be written as v = (5 m/s2) t + 2 m/s. See the difference? You need to use proper variables and units in order to compare it to theory or make actual conclusions about physical principles. The second equation tells me that when the data collection began (t = 0), the velocity of the object was 2 m/s. It also tells me that the velocity was changing at 5 m/s every second (m/s/s = m/s2). Let’s practice this a little, shall we? Force vs. mass F (N) y = 6.4x + 0.3 m (kg) You’ve just done a lab to see how much force was necessary to get a mass moving along a rough surface. Excel spat out the graph above. You labeled each axis with the variable and units (well done!). You titled the graph starting with the variable on the y-axis (nice job!). Now we turn to the equation. First we replace x and y with the variables we actually graphed: F = 6.4m + 0.3 Then we add units to our slope and intercept. The slope is found by dividing the rise by the run so the units will be the units from the y-axis divided by the units from the x-axis. -
History of Mathematics
Georgia Department of Education History of Mathematics K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by using manipulatives and a variety of representations, working independently and cooperatively to solve problems, estimating and computing efficiently, and conducting investigations and recording findings. There is a shift towards applying mathematical concepts and skills in the context of authentic problems and for the student to understand concepts rather than merely follow a sequence of procedures. In mathematics classrooms, students will learn to think critically in a mathematical way with an understanding that there are many different ways to a solution and sometimes more than one right answer in applied mathematics. Mathematics is the economy of information. The central idea of all mathematics is to discover how knowing some things well, via reasoning, permit students to know much else—without having to commit the information to memory as a separate fact. It is the reasoned, logical connections that make mathematics coherent. The implementation of the Georgia Standards of Excellence in Mathematics places a greater emphasis on sense making, problem solving, reasoning, representation, connections, and communication. History of Mathematics History of Mathematics is a one-semester elective course option for students who have completed AP Calculus or are taking AP Calculus concurrently. It traces the development of major branches of mathematics throughout history, specifically algebra, geometry, number theory, and methods of proofs, how that development was influenced by the needs of various cultures, and how the mathematics in turn influenced culture. The course extends the numbers and counting, algebra, geometry, and data analysis and probability strands from previous courses, and includes a new history strand. -
And Are Congruent Chords, So the Corresponding Arcs RS and ST Are Congruent
9-3 Arcs and Chords ALGEBRA Find the value of x. 3. SOLUTION: 1. In the same circle or in congruent circles, two minor SOLUTION: arcs are congruent if and only if their corresponding Arc ST is a minor arc, so m(arc ST) is equal to the chords are congruent. Since m(arc AB) = m(arc CD) measure of its related central angle or 93. = 127, arc AB arc CD and . and are congruent chords, so the corresponding arcs RS and ST are congruent. m(arc RS) = m(arc ST) and by substitution, x = 93. ANSWER: 93 ANSWER: 3 In , JK = 10 and . Find each measure. Round to the nearest hundredth. 2. SOLUTION: Since HG = 4 and FG = 4, and are 4. congruent chords and the corresponding arcs HG and FG are congruent. SOLUTION: m(arc HG) = m(arc FG) = x Radius is perpendicular to chord . So, by Arc HG, arc GF, and arc FH are adjacent arcs that Theorem 10.3, bisects arc JKL. Therefore, m(arc form the circle, so the sum of their measures is 360. JL) = m(arc LK). By substitution, m(arc JL) = or 67. ANSWER: 67 ANSWER: 70 eSolutions Manual - Powered by Cognero Page 1 9-3 Arcs and Chords 5. PQ ALGEBRA Find the value of x. SOLUTION: Draw radius and create right triangle PJQ. PM = 6 and since all radii of a circle are congruent, PJ = 6. Since the radius is perpendicular to , bisects by Theorem 10.3. So, JQ = (10) or 5. 7. Use the Pythagorean Theorem to find PQ. -
The Geometry of the Phase Diffusion Equation
J. Nonlinear Sci. Vol. 10: pp. 223–274 (2000) DOI: 10.1007/s003329910010 © 2000 Springer-Verlag New York Inc. The Geometry of the Phase Diffusion Equation N. M. Ercolani,1 R. Indik,1 A. C. Newell,1,2 and T. Passot3 1 Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA 2 Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK 3 CNRS UMR 6529, Observatoire de la Cˆote d’Azur, 06304 Nice Cedex 4, France Received on October 30, 1998; final revision received July 6, 1999 Communicated by Robert Kohn E E Summary. The Cross-Newell phase diffusion equation, (|k|)2T =∇(B(|k|) kE), kE =∇2, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order “self-dual” equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. 1. Introduction The mathematical models discussed in this paper are motivated by physical systems, far from equilibrium, which spontaneously form patterns. -
Squaring the Circle in Elliptic Geometry
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 2 Article 1 Squaring the Circle in Elliptic Geometry Noah Davis Aquinas College Kyle Jansens Aquinas College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Davis, Noah and Jansens, Kyle (2017) "Squaring the Circle in Elliptic Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 2 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss2/1 Rose- Hulman Undergraduate Mathematics Journal squaring the circle in elliptic geometry Noah Davis a Kyle Jansensb Volume 18, No. 2, Fall 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a [email protected] Aquinas College b scholar.rose-hulman.edu/rhumj Aquinas College Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 2, Fall 2017 squaring the circle in elliptic geometry Noah Davis Kyle Jansens Abstract. Constructing a regular quadrilateral (square) and circle of equal area was proved impossible in Euclidean geometry in 1882. Hyperbolic geometry, however, allows this construction. In this article, we complete the story, providing and proving a construction for squaring the circle in elliptic geometry. We also find the same additional requirements as the hyperbolic case: only certain angle sizes work for the squares and only certain radius sizes work for the circles; and the square and circle constructions do not rely on each other. Acknowledgements: We thank the Mohler-Thompson Program for supporting our work in summer 2014. Page 2 RHIT Undergrad. Math. J., Vol. 18, No. 2 1 Introduction In the Rose-Hulman Undergraduate Math Journal, 15 1 2014, Noah Davis demonstrated the construction of a hyperbolic circle and hyperbolic square in the Poincar´edisk [1]. -
Numerical Algebraic Geometry and Algebraic Kinematics
Numerical Algebraic Geometry and Algebraic Kinematics Charles W. Wampler∗ Andrew J. Sommese† January 14, 2011 Abstract In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work. Contents 1 Introduction 4 2 Notation 5 I Fundamentals of Algebraic Kinematics 6 3 Some Motivating Examples 6 3.1 Serial-LinkRobots ............................... ..... 6 3.1.1 Planar3Rrobot ................................. 6 3.1.2 Spatial6Rrobot ................................ 11 3.2 Four-BarLinkages ................................ 13 3.3 PlatformRobots .................................. 18 ∗General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090- 9055, U.S.A. Email: [email protected] URL: www.nd.edu/˜cwample1. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and by General Motors Research and Development. †Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, U.S.A. Email: [email protected] URL: www.nd.edu/˜sommese. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and the Duncan Chair of the University of Notre Dame. -
10-1 Study Guide and Intervention Circles and Circumference
NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 10-1 Study Guide and Intervention Circles and Circumference Segments in Circles A circle consists of all points in a plane that are a given distance, called the radius, from a given point called the center. A segment or line can intersect a circle in several ways. • A segment with endpoints that are at the center and on the circle is a radius. • A segment with endpoints on the circle is a chord. • A chord that passes through the circle’s center and made up of collinear radii is a diameter. chord: 퐴퐸̅̅̅̅, 퐵퐷̅̅̅̅ radius: 퐹퐵̅̅̅̅, 퐹퐶̅̅̅̅, 퐹퐷̅̅̅̅ For a circle that has radius r and diameter d, the following are true diameter: 퐵퐷̅̅̅̅ 푑 1 r = r = d d = 2r 2 2 Example a. Name the circle. The name of the circle is ⨀O. b. Name radii of the circle. 퐴푂̅̅̅̅, 퐵푂̅̅̅̅, 퐶푂̅̅̅̅, and 퐷푂̅̅̅̅ are radii. c. Name chords of the circle. 퐴퐵̅̅̅̅ and 퐶퐷̅̅̅̅ are chords. Exercises For Exercises 1-7, refer to 1. Name the circle. ⨀R 2. Name radii of the circle. 푹푨̅̅̅̅, 푹푩̅̅̅̅, 푹풀̅̅̅̅, and 푹푿̅̅̅̅ 3. Name chords of the circle. ̅푩풀̅̅̅, 푨푿̅̅̅̅, 푨푩̅̅̅̅, and 푿풀̅̅̅̅ 4. Name diameters of the circle. 푨푩̅̅̅̅, and 푿풀̅̅̅̅ 5. If AB = 18 millimeters, find AR. 9 mm 6. If RY = 10 inches, find AR and AB . AR = 10 in.; AB = 20 in. 7. Is 퐴퐵̅̅̅̅ ≅ 푋푌̅̅̅̅? Explain. Yes; all diameters of the same circle are congruent. Chapter 10 5 Glencoe Geometry NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 10-1 Study Guide and Intervention (continued) Circles and Circumference Circumference The circumference of a circle is the distance around the circle.