MSG.09.Plane Geometry.Pdf
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Chap 3-Triangles.Pdf
Defining Triangles Name(s): In this lesson, you'll experiment with an ordinary triangle and with special triangles that were constructed with constraints. The constraints limit what you can change in the triangie when you drag so that certain reiationships among angles and sides always hold. By observing these relationships, you will classify the triangles. 1. Open the sketch Classify Triangles.gsp. BD 2. Drag different vertices of h--.- \ each of the four triangles / \ t\ to observe and compare AL___\c yle how the triangles behave. QI Which of the kiangles LIqA seems the most flexible? ------------__. \ ,/ \ Explain. * M\----"--: "<-----\U \ a2 \Alhich of the triangles seems the least flexible? Explain. -: measure each I - :-f le, select three l? 3. Measure the three angles in AABC. coints, with the rei€x I loUr middle I 4. Draga vertex of LABC and observe the changing angle measures. To Then, in =ection. I le r/easure menu, answer the following questions, note how each angle can change from I :hoose Angle. I being acute to being right or obtuse. Q3 How many acute angles can a triangle have? ;;,i:?:PJi;il: I qo How many obtuse angles can a triangle have? to classify -sed I rc:s. These terms f> Qg LABC can be an obtuse triangle or an acute :"a- arso be used to I : assify triangles I triangle. One other triangle in the sketch can according to I also be either acute or obtuse. Which triangle is it? their angles. I aG Which triangle is always a right triangle, no matter what you drag? 07 Which triangle is always an equiangular triangle? -: -Fisure a tensth, I =ect a seoment. -
Holley GM LS Race Single-Plane Intake Manifold Kits
Holley GM LS Race Single-Plane Intake Manifold Kits 300-255 / 300-255BK LS1/2/6 Port-EFI - w/ Fuel Rails 4150 Flange 300-256 / 300-256BK LS1/2/6 Carbureted/TB EFI 4150 Flange 300-290 / 300-290BK LS3/L92 Port-EFI - w/ Fuel Rails 4150 Flange 300-291 / 300-291BK LS3/L92 Carbureted/TB EFI 4150 Flange 300-294 / 300-294BK LS1/2/6 Port-EFI - w/ Fuel Rails 4500 Flange 300-295 / 300-295BK LS1/2/6 Carbureted/TB EFI 4500 Flange IMPORTANT: Before installation, please read these instructions completely. APPLICATIONS: The Holley LS Race single-plane intake manifolds are designed for GM LS Gen III and IV engines, utilized in numerous performance applications, and are intended for carbureted, throttle body EFI, or direct-port EFI setups. The LS Race single-plane intake manifolds are designed for hi-performance/racing engine applications, 5.3 to 6.2+ liter displacement, and maximum engine speeds of 6000-7000 rpm, depending on the engine combination. This single-plane design provides optimal performance across the RPM spectrum while providing maximum performance up to 7000 rpm. These intake manifolds are for use on non-emissions controlled applications only, and will not accept stock components and hardware. Port EFI versions may not be compatible with all throttle body linkages. When installing the throttle body, make certain there is a minimum of ¼” clearance between all linkage and the fuel rail. SPLIT DESIGN: The Holley LS Race manifold incorporates a split feature, which allows disassembly of the intake for direct access to internal plenum and port surfaces, making custom porting and matching a snap. -
Pumαc - Power Round
PUMαC - Power Round. Geometry Revisited Stobaeus (one of Euclid's students): "But what shall I get by-learning these things?" Euclid to his slave: "Give him three pence, since he must make gain out of what he learns." - Euclid, Elements As the title puts it, this is going to be a geometry power round. And as it always happens with geometry problems, every single result involves certain particular con- figurations that have been studied and re-studied multiple times in literature. So even though we tryed to make it as self-contained as possible, there are a few basic preliminary things you should keep in mind when thinking about these problems. Of course, you might find solutions separate from these ideas, however, it would be unfair not to give a general overview of most of these tools that we ourselves used when we found these results. In any case, feel free to skip the next section and start working on the test if you know these things or feel that such a section might bound your spectrum of ideas or anything! 1 Prerequisites Exercise -6 (1 point). Let ABC be a triangle and let A1 2 BC, B1 2 AC, C1 2 AB, with none of A1, B1, and C1 a vertex of ABC. When we say that A1 2 BC, B1 2 AC, C1 2 AB, we mean that A1 is in the line passing through BC, not necessarily the line segment between them. Prove that if A1, B1, and C1 are collinear, then either none or exactly two of A1, B1, and C1 are in the corresponding line segments. -
A Historical Introduction to Elementary Geometry
i MATH 119 – Fall 2012: A HISTORICAL INTRODUCTION TO ELEMENTARY GEOMETRY Geometry is an word derived from ancient Greek meaning “earth measure” ( ge = earth or land ) + ( metria = measure ) . Euclid wrote the Elements of geometry between 330 and 320 B.C. It was a compilation of the major theorems on plane and solid geometry presented in an axiomatic style. Near the beginning of the first of the thirteen books of the Elements, Euclid enumerated five fundamental assumptions called postulates or axioms which he used to prove many related propositions or theorems on the geometry of two and three dimensions. POSTULATE 1. Any two points can be joined by a straight line. POSTULATE 2. Any straight line segment can be extended indefinitely in a straight line. POSTULATE 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. POSTULATE 4. All right angles are congruent. POSTULATE 5. (Parallel postulate) If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate 3 defines a sphere. Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom, which also holds only in the plane: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. -
Geometry Course Outline
GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid -
Geometry (Lines) SOL 4.14?
Geometry (Lines) lines, segments, rays, points, angles, intersecting, parallel, & perpendicular Point “A point is an exact location in space. It has no length or width.” Points have names; represented with a Capital Letter. – Example: A a Lines “A line is a collection of points going on and on infinitely in both directions. It has no endpoints.” A straight line that continues forever It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified because it has arrows on the ends. It is named by “a single lower case letter”. – Example: “line a” A B C D Line Segment “A line segment is part of a line. It has two endpoints and includes all the points between those endpoints. “ A straight line that stops It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified by points at the ends It is named by the Capital Letter End Points – Example: “line segment AB” or “line segment AD” C B A Ray “A ray is part of a line. It has one endpoint and continues on and on in one direction.” A straight line that stops on one end and keeps going on the other. It can go – Vertically – Horizontally – Obliquely (diagonally) It is identified by a point at one end and an arrow at the other. It can be named by saying the endpoint first and then say the name of one other point on the ray. – Example: “Ray AC” or “Ray AB” C Angles A B Two Rays That Have the Same Endpoint Form an Angle. This Endpoint Is Called the Vertex. -
Unsteadiness in Flow Over a Flat Plate at Angle-Of-Attack at Low Reynolds Numbers∗
Unsteadiness in Flow over a Flat Plate at Angle-of-Attack at Low Reynolds Numbers∗ Kunihiko Taira,† William B. Dickson,‡ Tim Colonius,§ Michael H. Dickinson¶ California Institute of Technology, Pasadena, California, 91125 Clarence W. Rowleyk Princeton University, Princeton, New Jersey, 08544 Flow over an impulsively started low-aspect-ratio flat plate at angle-of-attack is inves- tigated for a Reynolds number of 300. Numerical simulations, validated by a companion experiment, are performed to study the influence of aspect ratio, angle of attack, and plan- form geometry on the interaction of the leading-edge and tip vortices and resulting lift and drag coefficients. Aspect ratio is found to significantly influence the wake pattern and the force experienced by the plate. For large aspect ratio plates, leading-edge vortices evolved into hairpin vortices that eventually detached from the plate, interacting with the tip vor- tices in a complex manner. Separation of the leading-edge vortex is delayed to some extent by having convective transport of the spanwise vorticity as observed in flow over elliptic, semicircular, and delta-shaped planforms. The time at which lift achieves its maximum is observed to be fairly constant over different aspect ratios, angles of attack, and planform geometries during the initial transient. Preliminary results are also presented for flow over plates with steady actuation near the leading edge. I. Introduction In recent years, flows around flapping insect wings have been investigated and, in particular, it has been observed that the leading-edge vortex (LEV) formed during stroke reversal remains stably attached throughout the wing stroke, greatly enhancing lift.1, 2 Such LEVs are found to be stable over a wide range of angles of attack and for Reynolds number over a range of Re ≈ O(102) − O(104). -
Points, Lines, and Planes a Point Is a Position in Space. a Point Has No Length Or Width Or Thickness
Points, Lines, and Planes A Point is a position in space. A point has no length or width or thickness. A point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. A A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. It does not have a beginning or end. A B C D A line consists of infinitely many points. The four points A, B, C, D are all on the same line. Postulate: Two points determine a line. We name a line by using any two points on the line, so the above line can be named as any of the following: ! ! ! ! ! AB BC AC AD CD Any three or more points that are on the same line are called colinear points. In the above, points A; B; C; D are all colinear. A Ray is part of a line that has a beginning point, and extends indefinitely to one direction. A B C D A ray is named by using its beginning point with another point it contains. −! −! −−! −−! In the above, ray AB is the same ray as AC or AD. But ray BD is not the same −−! ray as AD. A (line) segment is a finite part of a line between two points, called its end points. A segment has a finite length. A B C D B C In the above, segment AD is not the same as segment BC Segment Addition Postulate: In a line segment, if points A; B; C are colinear and point B is between point A and point C, then AB + BC = AC You may look at the plus sign, +, as adding the length of the segments as numbers. -
The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes
S S symmetry Article The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes Mutahir Ali *, Farhad Ali, Abdus Saboor, M. Saad Ghafar and Amir Sultan Khan Department of Mathematics, Kohat University of Science and Technology, Kohat 26000, Khyber Pakhtunkhwa, Pakistan; [email protected] (F.A.); [email protected] (A.S.); [email protected] (M.S.G.); [email protected] (A.S.K.) * Correspondence: [email protected] Received: 5 December 2018; Accepted: 22 January 2019; Published: 13 February 2019 Abstract: This research provides second-order approximate Noether symmetries of geodetic Lagrangian of time-conformal plane symmetric spacetime. A time-conformal factor is of the form ee f (t) which perturbs the plane symmetric static spacetime, where e is small a positive parameter that produces perturbation in the spacetime. By considering the perturbation up to second-order in e in plane symmetric spacetime, we find the second order approximate Noether symmetries for the corresponding Lagrangian. Using Noether theorem, the corresponding second order approximate conservation laws are investigated for plane symmetric gravitational waves like spacetimes. This technique tells about the energy content of the gravitational waves. Keywords: Einstein field equations; time conformal spacetime; approximate conservation of energy 1. Introduction Gravitational waves are ripples in the fabric of space-time produced by some of the most violent and energetic processes like colliding black holes or closely orbiting black holes and neutron stars (binary pulsars). These waves travel with the speed of light and depend on their sources [1–5]. The study of these waves provide us useful information about their sources (black holes and neutron stars). -
The Mass of an Asymptotically Flat Manifold
The Mass of an Asymptotically Flat Manifold ROBERT BARTNIK Australian National University Abstract We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic operators with rough coefficients on weighted Sobolev spaces are summarised. Some relations between the mass. xalar curvature and harmonic maps are described and the positive mass theorem for ,c-dimensional spin manifolds is proved. Introduction Suppose that (M,g) is an asymptotically flat 3-manifold. In general relativity the mass of M is given by where glJ, denotes the partial derivative and dS‘ is the normal surface element to S,, the sphere at infinity. This expression is generally attributed to [3]; for a recent review of this and other expressions for the mass and other physically interesting quantities see (41. However, in all these works the definition depends on the choice of coordinates at infinity and it is certainly not clear whether or not this dependence is spurious. Since it is physically quite reasonable to assume that a frame at infinity (“observer”) is given, this point has not received much attention in the literature (but see [15]). It is our purpose in this paper to show that, under appropriate decay conditions on the metric, (0.1) generalises to n-dimensions for n 2 3 and gives an invariant of the metric structure (M,g). The decay conditions roughly stated (for n = 3) are lg - 61 = o(r-’12), lag1 = o(f312), etc, and thus include the usual falloff conditions. -
High School Geometry This Course Covers the Topics Shown Below
High School Geometry This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Curriculum Arithmetic and Algebra Review (151 topics) Fractions and Decimals (28 topics) Factors Greatest common factor of 2 numbers Equivalent fractions Simplifying a fraction Division involving zero Introduction to addition or subtraction of fractions with different denominators Addition or subtraction of fractions with different denominators Product of a unit fraction and a whole number Product of a fraction and a whole number: Problem type 1 Fraction multiplication Product of a fraction and a whole number: Problem type 2 The reciprocal of a number Division involving a whole number and a fraction Fraction division Complex fraction without variables: Problem type 1 Decimal place value: Tenths and hundredths Rounding decimals Introduction to ordering decimals Using a calculator to convert a fraction to a rounded decimal Addition of aligned decimals Decimal subtraction: Basic Decimal subtraction: Advanced Word problem with addition or subtraction of 2 decimals Multiplication of a decimal by a power of ten Introduction to decimal multiplication Multiplying a decimal by a whole number Word problem with multiple decimal operations: Problem type 1 Converting a fraction to a terminating decimal: Basic Signed Numbers (14 topics) Plotting integers on a number line Ordering integers Absolute value of a number Integer addition: Problem type 1 Integer addition: Problem type 2 Integer subtraction: Problem type 1 Integer -
MATH32052 Hyperbolic Geometry
MATH32052 Hyperbolic Geometry Charles Walkden 12th January, 2019 MATH32052 Contents Contents 0 Preliminaries 3 1 Where we are going 6 2 Length and distance in hyperbolic geometry 13 3 Circles and lines, M¨obius transformations 18 4 M¨obius transformations and geodesics in H 23 5 More on the geodesics in H 26 6 The Poincar´edisc model 39 7 The Gauss-Bonnet Theorem 44 8 Hyperbolic triangles 52 9 Fixed points of M¨obius transformations 56 10 Classifying M¨obius transformations: conjugacy, trace, and applications to parabolic transformations 59 11 Classifying M¨obius transformations: hyperbolic and elliptic transforma- tions 62 12 Fuchsian groups 66 13 Fundamental domains 71 14 Dirichlet polygons: the construction 75 15 Dirichlet polygons: examples 79 16 Side-pairing transformations 84 17 Elliptic cycles 87 18 Generators and relations 92 19 Poincar´e’s Theorem: the case of no boundary vertices 97 20 Poincar´e’s Theorem: the case of boundary vertices 102 c The University of Manchester 1 MATH32052 Contents 21 The signature of a Fuchsian group 109 22 Existence of a Fuchsian group with a given signature 117 23 Where we could go next 123 24 All of the exercises 126 25 Solutions 138 c The University of Manchester 2 MATH32052 0. Preliminaries 0. Preliminaries 0.1 Contact details § The lecturer is Dr Charles Walkden, Room 2.241, Tel: 0161 27 55805, Email: [email protected]. My office hour is: WHEN?. If you want to see me at another time then please email me first to arrange a mutually convenient time. 0.2 Course structure § 0.2.1 MATH32052 § MATH32052 Hyperbolic Geoemtry is a 10 credit course.