Schaum''s Outline of Geometry
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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 . -
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. -
P. 1 Math 490 Notes 7 Zero Dimensional Spaces for (SΩ,Τo)
p. 1 Math 490 Notes 7 Zero Dimensional Spaces For (SΩ, τo), discussed in our last set of notes, we can describe a basis B for τo as follows: B = {[λ, λ] ¯ λ is a non-limit ordinal } ∪ {[µ + 1, λ] ¯ λ is a limit ordinal and µ < λ}. ¯ ¯ The sets in B are τo-open, since they form a basis for the order topology, but they are also closed by the previous Prop N7.1 from our last set of notes. Sets which are simultaneously open and closed relative to the same topology are called clopen sets. A topology with a basis of clopen sets is defined to be zero-dimensional. As we have just discussed, (SΩ, τ0) is zero-dimensional, as are the discrete and indiscrete topologies on any set. It can also be shown that the Sorgenfrey line (R, τs) is zero-dimensional. Recall that a basis for τs is B = {[a, b) ¯ a, b ∈R and a < b}. It is easy to show that each set [a, b) is clopen relative to τs: ¯ each [a, b) itself is τs-open by definition of τs, and the complement of [a, b)is(−∞,a)∪[b, ∞), which can be written as [ ¡[a − n, a) ∪ [b, b + n)¢, and is therefore open. n∈N Closures and Interiors of Sets As you may know from studying analysis, subsets are frequently neither open nor closed. However, for any subset A in a topological space, there is a certain closed set A and a certain open set Ao associated with A in a natural way: Clτ A = A = \{B ¯ B is closed and A ⊆ B} (Closure of A) ¯ o Iτ A = A = [{U ¯ U is open and U ⊆ A}. -
The Geometry of Diagonal Groups
The geometry of diagonal groups R. A. Baileya, Peter J. Camerona, Cheryl E. Praegerb, Csaba Schneiderc aUniversity of St Andrews, St Andrews, UK bUniversity of Western Australia, Perth, Australia cUniversidade Federal de Minas Gerais, Belo Horizonte, Brazil Abstract Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan{Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combi- natorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such struct- ures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan{Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T . For m = 2, it is a Latin square, the Cayley table of T , though in fact any Latin square satisfies our combinatorial axioms. However, for m > 3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher- dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
Simple Polygons Scribe: Michael Goldwasser
CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons Scribe: Michael Goldwasser Algorithms for triangulating polygons are important tools throughout computa- tional geometry. Many problems involving polygons are simplified by partitioning the complex polygon into triangles, and then working with the individual triangles. The applications of such algorithms are well documented in papers involving visibility, motion planning, and computer graphics. The following notes give an introduction to triangulations and many related definitions and basic lemmas. Most of the definitions are based on a simple polygon, P, containing n edges, and hence n vertices. However, many of the definitions and results can be extended to a general arrangement of n line segments. 1 Diagonals Definition 1. Given a simple polygon, P, a diagonal is a line segment between two non-adjacent vertices that lies entirely within the interior of the polygon. Lemma 2. Every simple polygon with jPj > 3 contains a diagonal. Proof: Consider some vertex v. If v has a diagonal, it’s party time. If not then the only vertices visible from v are its neighbors. Therefore v must see some single edge beyond its neighbors that entirely spans the sector of visibility, and therefore v must be a convex vertex. Now consider the two neighbors of v. Since jPj > 3, these cannot be neighbors of each other, however they must be visible from each other because of the above situation, and thus the segment connecting them is indeed a diagonal. -
The Motion of Point Particles in Curved Spacetime
Living Rev. Relativity, 14, (2011), 7 LIVINGREVIEWS http://www.livingreviews.org/lrr-2011-7 (Update of lrr-2004-6) in relativity The Motion of Point Particles in Curved Spacetime Eric Poisson Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 email: [email protected] http://www.physics.uoguelph.ca/ Adam Pound Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 email: [email protected] Ian Vega Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 email: [email protected] Accepted on 23 August 2011 Published on 29 September 2011 Abstract This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise toa self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle. The field’s action on the particle is difficult to calculate because of its singular nature:the field diverges at the position of the particle. But it is possible to isolate the field’s singular part and show that it exerts no force on the particle { its only effect is to contribute to the particle's inertia. -
Unit 6 Visualising Solid Shapes(Final)
• 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure. -
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 -
My Favourite Problem No.1 Solution
My Favourite Problem No.1 Solution First write on the measurements given. The shape can be split into different sections as shown: d 8 e c b a 10 From Pythagoras’ Theorem we can see that BC2 = AC2 + AB2 100 = 64 + AB2 AB = 6 Area a + b + c = Area of Semi-circle on BC (radius 5) = Area c + d = Area of Semi-circle on AC (radius 4) = Area b + e = Area of Semi-circle on AB (radius 3) = Shaded area = Sum of semi-circles on AC and AB – Semi-circle on BC + Triangle Area d + e = d + c + b + e - (a + b + c ) + a = + - + = = 24 (i.e. shaded area is equal to the area of the triangle) The final solution required is one third of this area = 8 Surprisingly you do not need to calculate the areas of the semi-circles as we can extend the use of Pythagoras’ Theorem for other shapes on the three sides of a right-angled triangle. Triangle ABC is a right-angled triangle, so the sum of the areas of the two smaller semi-circles is equal to the area of the larger semi-circle on the hypotenuse BC. Consider a right- angled triangle of sides a, b and c. Then from Pythagoras’ Theorem we have that the square on the hypotenuse is equal to c a the sum of the squares on the other two sides. b Now if you multiply both sides by , this gives c which rearranges to . a b This can be interpreted as the area of the semi-circle on the hypotenuse is equal to the sum of the areas of the semi-circles on the other two sides. -
Name: Introduction to Geometry: Points, Lines and Planes Euclid
Name: Introduction to Geometry: Points, Lines and Planes Euclid: “What’s the point of Geometry?- Euclid” http://www.youtube.com/watch?v=_KUGLOiZyK8&safe=active When do historians believe that Euclid completed his work? What key topic did Euclid build on to create all of Geometry? What is an axiom? Why do you think that Euclid’s ideas have lasted for thousands of years? *Point – is a ________ in space. A point has ___ dimension. A point is name by a single capital letter. (ex) *Line – consists of an __________ number of points which extend in ___________ directions. A line is ___-dimensional. A line may be named by using a single lower case letter OR by any two points on the line. (ex) *Plane – consists of an __________ number of points which form a flat surface that extends in all directions. A plane is _____-dimensional. A plane may be named using a single capital letter OR by using three non- collinear points. (ex) *Line segment – consists of ____ points on a line, called the endpoints and all of the points between them. (ex) *Ray – consists of ___ point on a line (called an endpoint or the initial point) and all of the points on one side of the point. (ex) *Opposite rays – share the same initial point and extend in opposite directions on the same line. *Collinear points – (ex) *Coplanar points – (ex) *Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of elements that the figures have in common.