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Proofs with Perpendicular Lines
3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units. -
Some Intersection Theorems for Ordered Sets and Graphs
IOURNAL OF COMBINATORIAL THEORY, Series A 43, 23-37 (1986) Some Intersection Theorems for Ordered Sets and Graphs F. R. K. CHUNG* AND R. L. GRAHAM AT&T Bell Laboratories, Murray Hill, New Jersey 07974 and *Bell Communications Research, Morristown, New Jersey P. FRANKL C.N.R.S., Paris, France AND J. B. SHEARER' Universify of California, Berkeley, California Communicated by the Managing Editors Received May 22, 1984 A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F of any F and F’ in F are: (i) FnF’# 0, where all FEF have k elements (Erdos, Ko, and Rado (1961)). (ii) IFn F’I > j (Katona (1964)). In this paper, we consider the following general question: For a given family B of subsets of [n] = { 1, 2,..., n}, what is the largest family F of subsets of [n] satsifying F,F’EF-FnFzB for some BE B. Of particular interest are those B for which the maximum families consist of so- called “kernel systems,” i.e., the family of all supersets of some fixed set in B. For example, we show that the set of all (cyclic) translates of a block of consecutive integers in [n] is such a family. It turns out rather unexpectedly that many of the results we obtain here depend strongly on properties of the well-known entropy function (from information theory). -
Rural Expressway Intersection Synthesis of Practice and Crash Analysis
RURAL EXPRESSWAY INTERSECTION SYNTHESIS OF PRACTICE AND CRASH ANALYSIS Sponsored by the Iowa Department of Transportation (CTRE Project 03-157) Final Report October 2004 Disclaimer Notice The opinions, fi ndings, and conclusions expressed in this publication are those of the authors and not necessarily those of the Iowa Department of Transportation. The sponsor(s) assume no liability for the contents or use of the information contained in this document. This report does not constitute a standard, specifi cation, or regulation. The sponsor(s) do not endorse products or manufacturers. About CTRE/ISU The mission of the Center for Transportation Research and Education (CTRE) at Iowa State Uni- versity is to develop and implement innovative methods, materials, and technologies for improv- ing transportation effi ciency, safety, and reliability while improving the learning environment of students, faculty, and staff in transportation-related fi elds. Technical Report Documentation Page 1. Report No. 2. Government Accession No. 3. Recipient’s Catalog No. CTRE Project 03-157 4. Title and Subtitle 5. Report Date Rural Expressway Intersection Synthesis of Practice and Crash Analysis October 2004 6. Performing Organization Code 7. Author(s) 8. Performing Organization Report No. T. H. Maze, Neal R. Hawkins, and Garrett Burchett 9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) Center for Transportation Research and Education Iowa State University 11. Contract or Grant No. 2901 South Loop Drive, Suite 3100 Ames, IA 50010-8634 12. Sponsoring Organization Name and Address 13. Type of Report and Period Covered Iowa Department of Transportation Final Report 800 Lincoln Way 14. Sponsoring Agency Code Ames, IA 50010 15. -
Complete Intersection Dimension
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. LUCHEZAR L. AVRAMOV VESSELIN N. GASHAROV IRENA V. PEEVA Complete intersection dimension Publications mathématiques de l’I.H.É.S., tome 86 (1997), p. 67-114 <http://www.numdam.org/item?id=PMIHES_1997__86__67_0> © Publications mathématiques de l’I.H.É.S., 1997, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPLETE INTERSECTION DIMENSION by LUGHEZAR L. AVRAMOV, VESSELIN N. GASHAROV, and IRENA V. PEEVA (1) Abstract. A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI- dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions. CONTENTS Introduction ................................................................................ 67 1. Homological dimensions ................................................................... 70 2. Quantum regular sequences .............................................................. -
And Are Lines on Sphere B That Contain Point Q
11-5 Spherical Geometry Name each of the following on sphere B. 3. a triangle SOLUTION: are examples of triangles on sphere B. 1. two lines containing point Q SOLUTION: and are lines on sphere B that contain point Q. ANSWER: 4. two segments on the same great circle SOLUTION: are segments on the same great circle. ANSWER: and 2. a segment containing point L SOLUTION: is a segment on sphere B that contains point L. ANSWER: SPORTS Determine whether figure X on each of the spheres shown is a line in spherical geometry. 5. Refer to the image on Page 829. SOLUTION: Notice that figure X does not go through the pole of ANSWER: the sphere. Therefore, figure X is not a great circle and so not a line in spherical geometry. ANSWER: no eSolutions Manual - Powered by Cognero Page 1 11-5 Spherical Geometry 6. Refer to the image on Page 829. 8. Perpendicular lines intersect at one point. SOLUTION: SOLUTION: Notice that the figure X passes through the center of Perpendicular great circles intersect at two points. the ball and is a great circle, so it is a line in spherical geometry. ANSWER: yes ANSWER: PERSEVERANC Determine whether the Perpendicular great circles intersect at two points. following postulate or property of plane Euclidean geometry has a corresponding Name two lines containing point M, a segment statement in spherical geometry. If so, write the containing point S, and a triangle in each of the corresponding statement. If not, explain your following spheres. reasoning. 7. The points on any line or line segment can be put into one-to-one correspondence with real numbers. -
Chapter 5 Safety
5 Safety 5.1 Introduction 103 5.2 Conflicts 104 5.2.1 Vehicle conflicts 105 5.2.2 Pedestrian conflicts 108 5.2.3 Bicycle conflicts 110 5.3 Crash Statistics 111 5.3.1 Comparisons to previous intersection treatment 111 5.3.2 Collision types 113 5.3.3 Pedestrians 117 5.3.4 Bicyclists 120 5.4 Crash Prediction Models 122 5.5 References 125 Exhibit 5-1. Vehicle conflict points for “T” Intersections with single-lane approaches. 105 Exhibit 5-2. Vehicle conflict point comparison for intersections with single-lane approaches. 106 Exhibit 5-3. Improper lane-use conflicts in double-lane roundabouts. 107 Exhibit 5-4. Improper turn conflicts in double-lane roundabouts. 108 Exhibit 5-5. Vehicle-pedestrian conflicts at signalized intersections. 109 Exhibit 5-6. Vehicle-pedestrian conflicts at single-lane roundabouts. 109 Exhibit 5-7. Bicycle conflicts at conventional intersections (showing two left-turn options). 110 Exhibit 5-8. Bicycle conflicts at roundabouts. 111 Exhibit 5-9. Average annual crash frequencies at 11 U.S. intersections converted to roundabouts. 112 Exhibit 5-10. Mean crash reductions in various countries. 112 Exhibit 5-11. Reported proportions of major crash types at roundabouts. 113 Exhibit 5-12. Comparison of collision types at roundabouts. 114 Exhibit 5-13. Graphical depiction of collision types at roundabouts. 115 Exhibit 5-14. Crash percentage per type of user for urban roundabouts in 15 towns in western France. 116 Exhibit 5-15. British crash rates for pedestrians at roundabouts and signalized intersections. 117 Exhibit 5-16. Percentage reduction in the number of crashes by mode at 181 converted Dutch roundabouts. -
Intersection of Convex Objects in Two and Three Dimensions
Intersection of Convex Objects in Two and Three Dimensions B. CHAZELLE Yale University, New Haven, Connecticut AND D. P. DOBKIN Princeton University, Princeton, New Jersey Abstract. One of the basic geometric operations involves determining whether a pair of convex objects intersect. This problem is well understood in a model of computation in which the objects are given as input and their intersection is returned as output. For many applications, however, it may be assumed that the objects already exist within the computer and that the only output desired is a single piece of data giving a common point if the objects intersect or reporting no intersection if they are disjoint. For this problem, none of the previous lower bounds are valid and algorithms are proposed requiring sublinear time for their solution in two and three dimensions. Categories and Subject Descriptors: E.l [Data]: Data Structures; F.2.2 [Analysis of Algorithms]: Nonnumerical Algorithms and Problems General Terms: Algorithms, Theory, Verification Additional Key Words and Phrases: Convex sets, Fibonacci search, Intersection 1. Introduction This paper describes fast algorithms for testing the predicate, Do convex objects P and Q intersect? where an object is taken to be a line or a polygon in two dimensions or a plane or a polyhedron in three dimensions. The related problem Given convex objects P and Q, compute their intersection has been well studied, resulting in linear lower bounds and linear or quasi-linear upper bounds [2, 4, 11, 15-171. Lower bounds for this problem use arguments This research was supported in part by the National Science Foundation under grants MCS 79-03428, MCS 81-14207, MCS 83-03925, and MCS 83-03926, and by the Defense Advanced Project Agency under contract F33615-78-C-1551, monitored by the Air Force Offtce of Scientific Research. -
What Are the Advantages of Roundabouts?
What is a roundabout? A roundabout is an intersection where traffic travels around a Circulatory central island in a counter- Truck Apron Roadway clockwise direction. Vehicles entering or exiting the roundabout must yield to vehicles, bicyclists, and pedestrians. Figure 1 presents the elements of a roundabout. Yield Line Splitter Island Figure 1: Elements of a Roundabout What are the advantages of roundabouts? • Less Traffic Conflict: Figure 2 compares the conflict points between a conventional intersection and a modern roundabout. The lower number of conflict points translates to less potential for accidents. • Greater safety(1): Primarily achieved by slower speeds and elimination of left turns. Design elements of the roundabouts cause drivers to reduce their speeds. • Efficient traffic flow: Up to 50% increase in traffic capacity • Reduced Pollution and fuel usage: Less stops, shorter queues and no left turn storage. • Money saved: No signal equipment to install or maintain, plus savings in electricity use. • Community benefits: Traffic calming and enhanced aesthetics by landscaping. (1) Statistics published by the U.S. Dept. of transportation, Federal Highway Administration shows roundabouts to have the following advantages over conventional intersections: • 90% reduction in fatalities • 76% reduction in injuries • 35% reduction in pedestrian accidents. Signalized Intersection Roundabout Figure 2: Conflict Point Comparison How to Use a Roundabout Driving a car • Slow down as you approach the intersection. • Yield to pedestrians and bicyclists crossing the roadway. • Watch for signs and pavement markings. • Enter the roundabout if gap in traffic is sufficient. • Drive in a counter-clockwise direction around the roundabout until you reach your exit. Do not stop or pass other vehicles. -
Chapter 9 Intersections
2005 Intersections CHAPTER 9 INTERSECTIONS 9.0 INTRODUCTION Intersections are intended to operate with vehicles, pedestrians, and bicycles proceeding in many directions, often at the same time. At such locations, traffic movements on two or more facilities are required to occupy a common area. It is this unique characteristic of intersections, the repeated occurrence of conflicts, that is the basis for most intersection design standards, criteria, and proper operating procedures. An intersection is defined as the general area where two or more highways join or cross, including the roadway and roadside facilities for traffic movements within it. Each highway radiating from an intersection and forming part of it is an intersection leg. The common intersection of two highways crossing each other has four legs. It is not recommended that an intersection have more than four legs. An intersection is an important part of a highway system because, to a great extent, the efficiency, safety, speed, cost of operation, and capacity depend on its design. Each intersection involves through or cross-traffic movements on one or more of the highways concerned and may involve turning movements between these highways. These movements may be handled by various means, such as signals, signing, and channelization, depending on the type of intersection. 9.1 GENERAL DESIGN CONSIDERATIONS AND OBJECTIVES The main objective of intersection design is to reduce the potential conflicts between motor vehicles, bicycles, pedestrians, and facilities while facilitating the convenience, ease, and comfort of the people traversing the intersection. The design should be fitted closely to the natural transitional paths and operating characteristics of the users. -
Movingforward
FORWARD movingfAll 2010 A quarterly review of news and information about Pennsylvania local roads. When to Use Stop Signs in Alleys A Guide to Understanding the State’s Requirements Related to Traffic-Control Devices at Alley Intersections by Patrick Wright, Pennoni Associates When deciding whether to use stop signs and other An alley is considered a “highway” in the Vehicle traffic-control devices in alleys, municipalities Code because it is a “roadway open to the use of the should be familiar with two major issues. The first public.” Following this logic, the junction of an alley is whether traffic control is even required, and the with another highway (including another alley) is con- second is how to properly place the signs especially sidered an “intersection” under the Vehicle Code, and within the space constraints found in most alleys. thus crosswalks (whether marked or unmarked) exist. Understanding Alleys What Traffic-Control and Intersections Devices Are Required? Alleys are defined separately in both the Now that the definitions of alleys and intersec- Pennsylvania Vehicle Code (Title 75) and the tions have been clarified, the next step is to deter- Manual on Uniform Traffic Control Devices mine what traffic-control devices are required for (MUTCD). According to the Vehicle Code (Title alleys. As at any intersection, the Vehicle Code does 75, Section 102) as well as the MUTCD, an alley not necessarily require stop signs or other traffic-con- is “a street or highway intended to provide access to trol devices. Instead, the code has specific “rules of the rear or side of lots or buildings in urban districts the road” that govern driving behavior and the right- and not intended for the purpose of through of-way at intersections depending on the situation. -
Arlington County Pavement Marking Specifications
DEPARTMENT OF ENVIRONMENTAL SERVICES ARLINGTON COUNTY PAVEMENT MARKING SPECIFICATIONS MAY 2017 T-1.1 PAVEMENT MARKINGS Table of Contents 1. General ................................................................................................................................................ 2 2. Design Criteria ...................................................................................................................................... 3 3. Marking Plan Preparation ..................................................................................................................... 4 Exhibits ...................................................................................................................................................... 5 MK – 1 Typical Crosswalk ......................................................................................................................... 5 MK – 1a Typical Crosswalk Details .............................................................................................................. 6 MK – 2 Typical Cross Section ..................................................................................................................... 7 MK – 3 Typical Speed Hump Markings ...................................................................................................... 8 MK – 4 Typical Speed Table ...................................................................................................................... 9 MK – 4a Typical Speed Hump Details ....................................................................................................... -
Complex Intersection Bodies
COMPLEX INTERSECTION BODIES A. KOLDOBSKY, G. PAOURIS, AND M. ZYMONOPOULOU Abstract. We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's the- orem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures and the corresponding hyperplane inequal- ity for measures of complex intersection bodies. 1. Introduction The concept of an intersection body was introduced by Lutwak [37], as part of his dual Brunn-Minkowski theory. In particular, these bodies played an important role in the solution of the Busemann-Petty prob- lem. Many results on intersection bodies have appeared in recent years (see [10, 22, 34] and references there), but almost all of them apply to the real case. The goal of this paper is to extend the concept of an intersection body to the complex case. Let K and L be origin symmetric star bodies in Rn: Following [37], we say that K is the intersection body of L if the radius of K in every direction is equal to the volume of the central hyperplane section of L perpendicular to this direction, i.e. for every ξ 2 Sn−1; −1 ? kξkK = jL \ ξ j; (1) ? n where kxkK = minfa ≥ 0 : x 2 aKg, ξ = fx 2 R :(x; ξ) = 0g; and j · j stands for volume. By a theorem of Busemann [8] the intersection body of an origin symmetric convex body is also convex.