Areas of Polygons Inscribed in a Circle
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem (For CBSE, ICSE, IAS, NET, NRA 2022)
9/22/2021 Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem- FlexiPrep FlexiPrep Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem (For CBSE, ICSE, IAS, NET, NRA 2022) Get unlimited access to the best preparation resource for competitive exams : get questions, notes, tests, video lectures and more- for all subjects of your exam. A quadrilateral is a 4-sided polygon bounded by 4 finite line segments. The word ‘quadrilateral’ is composed of two Latin words, Quadric meaning ‘four’ and latus meaning ‘side’ . It is a two-dimensional figure having four sides (or edges) and four vertices. A circle is the locus of all points in a plane which are equidistant from a fixed point. If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle, then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined, they form vertices of a cyclic quadrilateral. It can be visualized as a quadrilateral which is inscribed in a circle, i.e.. all four vertices of the quadrilateral lie on the circumference of the circle. What is a Cyclic Quadrilateral? In the figure given below, the quadrilateral ABCD is cyclic. ©FlexiPrep. Report ©violations @https://tips.fbi.gov/ 1 of 5 9/22/2021 Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem- FlexiPrep Let us do an activity. Take a circle and choose any 4 points on the circumference of the circle. Join these points to form a quadrilateral. Now measure the angles formed at the vertices of the cyclic quadrilateral. -
Extremal Problems for Convex Polygons ∗
Extremal Problems for Convex Polygons ∗ Charles Audet Ecole´ Polytechnique de Montreal´ Pierre Hansen HEC Montreal´ Fred´ eric´ Messine ENSEEIHT-IRIT Abstract. Consider a convex polygon Vn with n sides, perimeter Pn, diameter Dn, area An, sum of distances between vertices Sn and width Wn. Minimizing or maximizing any of these quantities while fixing another defines ten pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization meth- ods. Numerous open problems are mentioned, as well as series of test problems for global optimization and nonlinear programming codes. Keywords: polygon, perimeter, diameter, area, sum of distances, width, isoperimeter problem, isodiametric problem. 1. Introduction Plane geometry is replete with extremal problems, many of which are de- scribed in the book of Croft, Falconer and Guy [12] on Unsolved problems in geometry. Traditionally, such problems have been solved, some since the Greeks, by geometrical reasoning. In the last four decades, this approach has been increasingly complemented by global optimization methods. This allowed solution of larger instances than could be solved by any one of these two approaches alone. Probably the best known type of such problems are circle packing ones: given a geometrical form such as a unit square, a unit-side triangle or a unit- diameter circle, find the maximum radius and configuration of n circles which can be packed in its interior (see [46] for a recent survey and the site [44] for a census of exact and approximate results with up to 300 circles). -
Ethnomathematics and Education in Africa
Copyright ©2014 by Paulus Gerdes www.lulu.com http://www.lulu.com/spotlight/pgerdes 2 Paulus Gerdes Second edition: ISTEG Belo Horizonte Boane Mozambique 2014 3 First Edition (January 1995): Institutionen för Internationell Pedagogik (Institute of International Education) Stockholms Universitet (University of Stockholm) Report 97 Second Edition (January 2014): Instituto Superior de Tecnologias e Gestão (ISTEG) (Higher Institute for Technology and Management) Av. de Namaacha 188, Belo Horizonte, Boane, Mozambique Distributed by: www.lulu.com http://www.lulu.com/spotlight/pgerdes Author: Paulus Gerdes African Academy of Sciences & ISTEG, Mozambique C.P. 915, Maputo, Mozambique ([email protected]) Photograph on the front cover: Detail of a Tonga basket acquired, in January 2014, by the author in Inhambane, Mozambique 4 CONTENTS page Preface (2014) 11 Chapter 1: Introduction 13 Chapter 2: Ethnomathematical research: preparing a 19 response to a major challenge to mathematics education in Africa Societal and educational background 19 A major challenge to mathematics education 21 Ethnomathematics Research Project in Mozambique 23 Chapter 3: On the concept of ethnomathematics 29 Ethnographers on ethnoscience 29 Genesis of the concept of ethnomathematics among 31 mathematicians and mathematics teachers Concept, accent or movement? 34 Bibliography 39 Chapter 4: How to recognize hidden geometrical thinking: 45 a contribution to the development of an anthropology of mathematics Confrontation 45 Introduction 46 First example 47 Second example -
ON a PROOF of the TREE CONJECTURE for TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich
ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich To cite this version: Olga Paris-Romaskevich. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. 2019. hal-02169195v1 HAL Id: hal-02169195 https://hal.archives-ouvertes.fr/hal-02169195v1 Preprint submitted on 30 Jun 2019 (v1), last revised 8 Oct 2019 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. OLGA PARIS-ROMASKEVICH To Manya and Katya, to the moments we shared around mathematics on one winter day in Moscow. Abstract. Tiling billiards model a movement of light in heterogeneous medium consisting of homogeneous cells in which the coefficient of refraction between two cells is equal to −1. The dynamics of such billiards depends strongly on the form of an underlying tiling. In this work we consider periodic tilings by triangles (and cyclic quadrilaterals), and define natural foliations associated to tiling billiards in these tilings. By studying these foliations we manage to prove the Tree Conjecture for triangle tiling billiards that was stated in the work by Baird-Smith, Davis, Fromm and Iyer, as well as its generalization that we call Density property. -
Circle and Cyclic Quadrilaterals
Circle and Cyclic Quadrilaterals MARIUS GHERGU School of Mathematics and Statistics University College Dublin 33 Basic Facts About Circles A central angle is an angle whose vertex is at the center of ² the circle. It measure is equal the measure of the intercepted arc. An angle whose vertex lies on the circle and legs intersect the ² cirlc is called inscribed in the circle. Its measure equals half length of the subtended arc of the circle. AOC= contral angle, AOC ÙAC \ \ Æ ABC=inscribed angle, ABC ÙAC \ \ Æ 2 A line that has exactly one common point with a circle is ² called tangent to the circle. 44 The tangent at a point A on a circle of is perpendicular to ² the diameter passing through A. OA AB ? Through a point A outside of a circle, exactly two tangent ² lines can be drawn. The two tangent segments drawn from an exterior point to a cricle are equal. OA OB, OBC OAC 90o ¢OAB ¢OBC Æ \ Æ \ Æ Æ) ´ 55 The value of the angle between chord AB and the tangent ² line to the circle that passes through A equals half the length of the arc AB. AC,BC chords CP tangent Æ Æ ÙAC ÙAC ABC , ACP \ Æ 2 \ Æ 2 The line passing through the centres of two tangent circles ² also contains their tangent point. 66 Cyclic Quadrilaterals A convex quadrilateral is called cyclic if its vertices lie on a ² circle. A convex quadrilateral is cyclic if and only if one of the fol- ² lowing equivalent conditions hold: (1) The sum of two opposite angles is 180o; (2) One angle formed by two consecutive sides of the quadri- lateral equal the external angle formed by the other two sides of the quadrilateral; (3) The angle between one side and a diagonal equals the angle between the opposite side and the other diagonal. -
Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry
Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry 16 Notes ANGLES IN A CIRCLE AND CYCLIC QUADRILATERAL You must have measured the angles between two straight lines. Let us now study the angles made by arcs and chords in a circle and a cyclic quadrilateral. OBJECTIVES After studying this lesson, you will be able to • verify that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle; • prove that angles in the same segment of a circle are equal; • cite examples of concyclic points; • define cyclic quadrilterals; • prove that sum of the opposite angles of a cyclic quadrilateral is 180o; • use properties of cyclic qudrilateral; • solve problems based on Theorems (proved) and solve other numerical problems based on verified properties; • use results of other theorems in solving problems. EXPECTED BACKGROUND KNOWLEDGE • Angles of a triangle • Arc, chord and circumference of a circle • Quadrilateral and its types Mathematics Secondary Course 395 MODULE - 3 Angles in a Circle and Cyclic Quadrilateral Geometry 16.1 ANGLES IN A CIRCLE Central Angle. The angle made at the centre of a circle by the radii at the end points of an arc (or Notes a chord) is called the central angle or angle subtended by an arc (or chord) at the centre. In Fig. 16.1, ∠POQ is the central angle made by arc PRQ. Fig. 16.1 The length of an arc is closely associated with the central angle subtended by the arc. Let us define the “degree measure” of an arc in terms of the central angle. -
Allowing Coverage Holes of Bounded Diameter in Wireless Sensor
1 Trap Coverage: Allowing Coverage Holes of Bounded Diameter in Wireless Sensor Networks Paul Balister x Zizhan Zhengy Santosh Kumarx Prasun Sinhay x University of Memphis y The Ohio State University fpbalistr,[email protected] fzhengz,[email protected] Abstract—Tracking of movements such as that of people, or blanket coverage [19]), as has been the case in prototype animals, vehicles, or of phenomena such as fire, can be achieved deployments [14], [17], [22]. The requirement of full coverage by deploying a wireless sensor network. So far only prototype will soon become a bottleneck as we begin to see real-life systems have been deployed and hence the issue of scale has not become critical. Real-life deployments, however, will be at large deployments. scale and achieving this scale will become prohibitively expensive In this paper, we therefore propose a new model of coverage, if we require every point in the region to be covered (i.e., full called Trap Coverage, that scales well with large deployment coverage), as has been the case in prototype deployments. regions. We define a Coverage Hole in a target region of In this paper we therefore propose a new model of coverage, deployment A to be a connected component1 of the set of called Trap Coverage, that scales well with large deployment regions. A sensor network providing Trap Coverage guarantees uncovered points of A. A sensor network is said to provide that any moving object or phenomena can move at most a Trap Coverage with diameter d to A if the diameter of any (known) displacement before it is guaranteed to be detected by Coverage Hole in A is at most d. -
Art Problem Solving
the ART of PROBLEM SOLVING Volume 1: the BASICS Sandor Lehoczky Richard Rusczyk Copyright © 1993,1995, 2003,2004, 2006 Sandor Lehoczky and Richard Rusczyk. All Rights Reserved. Reproduction of any part of this book without expressed permission of the authors is strictly forbid den. For use outside the classroom of the problems contained herein, permission must be acquired from the cited sources. ISBN-10: 0-9773045-6-6 ISBN-13: 978-0-9773045-6-1 Published by: AoPS Incorporated P.O. Box 2185 Alpine, CA 91903-2185 (619) 659-1612 [email protected] Visit the Art of Problem Solving website at http: //www. artofproblemsolving. com Printed in the United States of America. Seventh Edition; printed in 2006. Editor: David Patrick Cover image designed by Vanessa Rusczyk using KaleidoTile software. Cover Image: "Grand Canyon from South Rim" by Ansel Adams. No permissions required; National Archive photo 79-AAF-8. This book was produced using the KTgX document processing system. Diagrams created by Maria Monks using METRPOST. To Ameyalli, for laughter as clear as your lake, for spirit strong and serene as your skyline, for all that you have taught and learned in four winters. And to Mrs. Wendt, who is still by far the best teacher I ever had. —SL For my desert flower Vanessa. Told you we'd make it there eventually. —RR Special thanks to the following people who helped make this possible: William and Claire Devlin, Sandor L. and Julieanne G. Lehoczky, Steve and Ann Rubio, Richard and Claire Rusczyk, Stanley Rusczyk. Thanks A large number of individuals and organizations have helped make the ART of PROBLEM SOLVING possible. -
Cyclic Quadrilaterals — the Big Picture Yufei Zhao [email protected]
Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao [email protected] An important skill of an olympiad geometer is being able to recognize known configurations. Indeed, many geometry problems are built on a few common themes. In this lecture, we will explore one such configuration. 1 What Do These Problems Have in Common? 1. (IMO 1985) A circle with center O passes through the vertices A and C of triangle ABC and intersects segments AB and BC again at distinct points K and N, respectively. The circumcircles of triangles ABC and KBN intersects at exactly two distinct points B and M. ◦ Prove that \OMB = 90 . B M N K O A C 2. (Russia 1995; Romanian TST 1996; Iran 1997) Consider a circle with diameter AB and center O, and let C and D be two points on this circle. The line CD meets the line AB at a point M satisfying MB < MA and MD < MC. Let K be the point of intersection (different from ◦ O) of the circumcircles of triangles AOC and DOB. Show that \MKO = 90 . C D K M A O B 3. (USA TST 2007) Triangle ABC is inscribed in circle !. The tangent lines to ! at B and C meet at T . Point S lies on ray BC such that AS ? AT . Points B1 and C1 lies on ray ST (with C1 in between B1 and S) such that B1T = BT = C1T . Prove that triangles ABC and AB1C1 are similar to each other. 1 Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao A B S C C1 B1 T Although these geometric configurations may seem very different at first sight, they are actually very related. -
Cyclic Quadrilaterals
GI_PAGES19-42 3/13/03 7:02 PM Page 1 Cyclic Quadrilaterals Definition: Cyclic quadrilateral—a quadrilateral inscribed in a circle (Figure 1). Construct and Investigate: 1. Construct a circle on the Voyage™ 200 with Cabri screen, and label its center O. Using the Polygon tool, construct quadrilateral ABCD where A, B, C, and D are on circle O. By the definition given Figure 1 above, ABCD is a cyclic quadrilateral (Figure 1). Cyclic quadrilaterals have many interesting and surprising properties. Use the Voyage 200 with Cabri tools to investigate the properties of cyclic quadrilateral ABCD. See whether you can discover several relationships that appear to be true regardless of the size of the circle or the location of A, B, C, and D on the circle. 2. Measure the lengths of the sides and diagonals of quadrilateral ABCD. See whether you can discover a relationship that is always true of these six measurements for all cyclic quadrilaterals. This relationship has been known for 1800 years and is called Ptolemy’s Theorem after Alexandrian mathematician Claudius Ptolemaeus (A.D. 85 to 165). 3. Determine which quadrilaterals from the quadrilateral hierarchy can be cyclic quadrilaterals (Figure 2). 4. Over 1300 years ago, the Hindu mathematician Brahmagupta discovered that the area of a cyclic Figure 2 quadrilateral can be determined by the formula: A = (s – a)(s – b)(s – c)(s – d) where a, b, c, and d are the lengths of the sides of the a + b + c + d quadrilateral and s is the semiperimeter given by s = 2 . Using cyclic quadrilaterals, verify these relationships. -
Cyclic and Bicentric Quadrilaterals G
Cyclic and Bicentric Quadrilaterals G. T. Springer Email: [email protected] Hewlett-Packard Calculators and Educational Software Abstract. In this hands-on workshop, participants will use the HP Prime graphing calculator and its dynamic geometry app to explore some of the many properties of cyclic and bicentric quadrilaterals. The workshop will start with a brief introduction to the HP Prime and an overview of its features to get novice participants oriented. Participants will then use ready-to-hand constructions of cyclic and bicentric quadrilaterals to explore. Part 1: Cyclic Quadrilaterals The instructor will send you an HP Prime app called CyclicQuad for this part of the activity. A cyclic quadrilateral is a convex quadrilateral that has a circumscribed circle. 1. Press ! to open the App Library and select the CyclicQuad app. The construction consists DEGH, a cyclic quadrilateral circumscribed by circle A. 2. Tap and drag any of the points D, E, G, or H to change the quadrilateral. Which of the following can DEGH never be? • Square • Rhombus (non-square) • Rectangle (non-square) • Parallelogram (non-rhombus) • Isosceles trapezoid • Kite Just move the points of the quadrilateral around enough to convince yourself for each one. Notice HDE and HE are both inscribed angles that subtend the entirety of the circle; ≮ ≮ likewise with DHG and DEG. This leads us to a defining characteristic of cyclic ≮ ≮ quadrilaterals. Make a conjecture. A quadrilateral is cyclic if and only if… 3. Make DEGH into a kite, similar to that shown to the right. Tap segment HE and press E to select it. Now use U and D to move the diagonal vertically. -
Fun Problems in Geometry and Beyond
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 097, 21 pages Fun Problems in Geometry and Beyond Boris KHESIN †y and Serge TABACHNIKOV ‡z †y Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada E-mail: [email protected] URL: http://www.math.toronto.edu/khesin/ ‡z Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA E-mail: [email protected] URL: http://www.personal.psu.edu/sot2/ Received November 17, 2019; Published online December 11, 2019 https://doi.org/10.3842/SIGMA.2019.097 Abstract. We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend “treasure\treasure chest”chest" supplemen- ting the course weekday lecture notes. The problems and solutions are not original, while their relation to the course might be so. Key words: clocks; spot it!; hunters; parking; frames; tangents; algebra; geometry 2010 Mathematics Subject Classification: 00A08; 00A09 To Dmitry Borisovich Fuchs on the occasion of his 80th anniversary Áîëüøå çàäà÷, õîðîøèõ è ðàçíûõ! attributed to Winston Churchill (1918) This is a belated birthday present to Dmitry Borisovich Fuchs, a mathematician young at 1 heart.1 Many of these problems belong to the mathematical folklore. We did not attempt to trace them back to their origins (which might be impossible anyway), and the given references are not necessarily the first mentioning of these problems. 1 Problems 1. AA wall wall clock clock has has the the minute minute and and hour hour hands hands of of equal equal length.