101.1 CP Lines and Segments That Intersect a Circle.Notebook April 17, 2017 Chapter 10 Circles Part 1 - Lines and Segments That Intersect a Circle

Total Page：16

File Type：pdf, Size：1020Kb

101.1 CP Lines and Segments that Intersect a Circle.notebook April 17, 2017 Chapter 10 Circles Part 1 - Lines and Segments that Intersect a Circle In this lesson we will: Name and describe the lines and segments that intersect a circle. A chord is a segment whose endpoints are on the circle. > BA is a chord in circle O. A diameter is a chord that intersects the center of the circle. > BA is a diameter in circle P. A radius is a segment whose endpoints are the center of the circle and a point on the circle. > PA is a radius in circle P. A radius of a circle is half the length of the circle's diameter. 1 101.1 CP Lines and Segments that Intersect a Circle.notebook April 17, 2017 A tangent is a line or segment that intersects a circle at one point. The point of tangency is the point where the line or segment intersects the circle > line t is a tangent line to circle O and P is the point of tangency A common tangent is a line that is tangent to two circles. A secant is a line that intersects a circle at two points. The segment part of a secant contained inside the circle is a chord. > line AB is a secant in circle O > segment AB is a chord in circle O 2 101.1 CP Lines and Segments that Intersect a Circle.notebook April 17, 2017 Examples: Naming Lines and Segments that Intersect Circles Name each line or segment that intersects 8L. (a) Name 2 chords (b) Name a diameter (c) Name 3 radii (d) Name a tangent line and the point of tangency (e) Name a secant Name each line or segment that intersects 8A. (a) Name 3 chords (b) Name a diameter (c) Name 2 radii (d) Name a tangent line and the point of tangency (e) Name a secant 3 101.1 CP Lines and Segments that Intersect a Circle.notebook April 17, 2017 Examples: Identifying Lines and Segments that Intersect Circles Identify each line or segment that intersects 8J. (a) KG (b) CK (c) GH (d) point E (e) GH Identify each line or segment that intersects 8Q. (a) line s (b) QR (c) VT (d) VT (e) WR 4 101.1 CP Lines and Segments that Intersect a Circle.notebook April 17, 2017 QUIZ NEXT CLASS 5.

Copy Link

Recommended publications

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 .
[Show full text]
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.
[Show full text]
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
[Show full text]
Find the Radius Or Diameter of the Circle with the Given Dimensions. 2
8-1 Circumference Find the radius or diameter of the circle with the given dimensions. 2. d = 24 ft SOLUTION: The radius is 12 feet. ANSWER: 12 ft Find the circumference of the circle. Use 3.14 or for π. Round to the nearest tenth if necessary. 4. SOLUTION: The circumference of the circle is about 25.1 feet. ANSWER: 3.14 × 8 = 25.1 ft eSolutions Manual - Powered by Cognero Page 1 8-1 Circumference 6. SOLUTION: The circumference of the circle is about 22 miles. ANSWER: 8. The Belknap shield volcano is located in the Cascade Range in Oregon. The volcano is circular and has a diameter of 5 miles. What is the circumference of this volcano to the nearest tenth? SOLUTION: The circumference of the volcano is about 15.7 miles. ANSWER: 15.7 mi eSolutions Manual - Powered by Cognero Page 2 8-1 Circumference Copy and Solve Show your work on a separate piece of paper. Find the diameter given the circumference of the object. Use 3.14 for π. 10. a satellite dish with a circumference of 957.7 meters SOLUTION: The diameter of the satellite dish is 305 meters. ANSWER: 305 m 12. a nickel with a circumference of about 65.94 millimeters SOLUTION: The diameter of the nickel is 21 millimeters. ANSWER: 21 mm eSolutions Manual - Powered by Cognero Page 3 8-1 Circumference Find the distance around the figure. Use 3.14 for π. 14. SOLUTION: The distance around the figure is 5 + 5 or 10 feet plus of the circumference of a circle with a radius of 5 feet.
[Show full text]
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.
[Show full text]
Lesson 2: Circles, Chords, Diameters, and Their Relationships
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M5 GEOMETRY Lesson 2: Circles, Chords, Diameters, and Their Relationships Student Outcomes . Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct the perpendicular bisector of a line segment and draw conclusions about points on that bisector and the endpoints of the segment. They relate the construction to the theorem stating that any perpendicular bisector of a chord must pass through the center of the circle. Classwork Opening Exercise (4 minutes) Scaffolding: Post a diagram and display the steps to create a Opening Exercise perpendicular bisector used in Construct the perpendicular bisector of line segment below (as you did in Module 1). Lesson 4 of Module 1. �� Label the endpoints of the segment and . Draw circle with center and radius . Draw another line that bisects but is not perpendicular to it. Draw circle with center �� and radius . List one similarity and one difference�� between the two bisectors. . Label the points of Answers will vary. All points on the perpendicular bisector are equidistant from points and 46T . �� Points on the other bisector are not equidistant from points A and B. The perpendicular bisector intersection as and . meets at right angles. The other bisector meets at angles that are not congruent. Draw . �� ⃖��⃗ You may wish to recall for students the definition of equidistant: EQUIDISTANT:. A point is said to be equidistant from two different points and if = . Points and can be replaced in the definition above with other figures (lines, etc.) as long as the distance to those figures is given meaning first.
[Show full text]
Creating a Revolve
Always begin your creo session by selecting your working directory If you select working directory at the beginning, that folder will be the default location when you try to open files, as well as when you save files. It makes everything a lot easier if you select your working directory first. For help opening/saving/downloading files see basics, for help creating an extrude see basic solids. Creating a Revolve Open a new part file, name it something unique. Choose a plane to sketch on Go to sketch view (if you don’t know where that is see Basic Solids) Move your cursor so the centerline snaps to the horizontal line as shown above You may now begin your sketch I have just sketched a random shape Sketch Tips: ● Your shape MUST be closed ● If you didn’t put a centerline in, you will get radial instead of diameter dimensions (in general this is bad) ● Remember this is being revolved so you are only sketching the profile on one side of the center line. If you need to put in diameter dimensions: Click normal, click the line or point you want the dimension for, click the centerline, and click the same line/point that you clicked the first time AGAIN, and then middle mouse button where you want the dimension to be placed. After your sketch is done, click the checkmark to get out of sketch, then click the checkmark again to complete the revolve The part in the sketch will look like this: .
[Show full text]
Maths Module 7 Geometry
Maths Module 7 Geometry This module covers concepts such as: • two and three dimensional shapes • measurement • perimeter, area and volume • circles www.jcu.edu.au/students/learning-centre Module 7 Geometry 1. Introduction 2. Two Dimensional Shapes 3. Area and Perimeter 4. Circles 5. Three Dimensional Shapes 6. Measurement 7. Area and Volume (extension) 8. Answers 9. Helpful Websites 2 1. Introduction Geometry exists in the world around us both in manmade and natural form. This first section covers some aspects of plane geometry such as lines, points, angles and circles. A point marks a point, a position, yet is does not have a size. However, physically, when we draw a point it will have width, but as an abstract concept, it really represents a point in our imagination. A line also has no width; it will extend infinitely in both directions. When drawing a line, physically it has width, yet it is an abstract representation in our minds. Given two distinct points for a line to pass though – there will only ever be the one line. The points are generally referred to with upper case letters and the line with lower case. For instance we may draw line though points and : The line may be referred to as line or as line . The distance between and is called a line segment or interval. If we had two lines, there would be two possibilities. They would either meet at a single point, and they would form angles at the intersection. Or they would never meet, and then they would be parallel lines.
[Show full text]
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, Möbius transformations 18 4 Möbius transformations and geodesics in H 23 5 More on the geodesics in H 26 6 The Poincarédisc model 39 7 The Gauss-Bonnet Theorem 44 8 Hyperbolic triangles 52 9 Fixed points of Möbius transformations 56 10 Classifying Möbius transformations: conjugacy, trace, and applications to parabolic transformations 59 11 Classifying Möbius 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 Poincaré’s Theorem: the case of no boundary vertices 97 20 Poincaré’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.
[Show full text]
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31, 1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a ThisworkwassupportedinpartbyTheGeometryCenter,UniversityofMinnesota,anSTC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. This has not been true of the mathematicians and physicists of our generation.
[Show full text]
Diameter Or Radius of a Circle Given Circumference
Diameter or Radius of a Circle Given Circumference Jen Kershaw Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) AUTHOR Jen Kershaw To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-proﬁt organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2013 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Com- mons from time to time (the “CC License”), which is incorporated herein by this reference.
[Show full text]
The Hyperbolic Plane
The Hyperbolic Plane Rich Schwartz November 19, 2014 1 Euclid’s Postulates Hyperbolic geometry arose out of an attempt to understand Euclid’s fifth postulate. So, first I am going to discuss Euclid’s postulates. Here they are: 1. Given any two distinct points in the plane, there is a line through them. 2. Any line segment may be extended to a line. 3. Given any point and any radius, there is a circle with that radius cen- tered at that point. 4. All right angles are equal to one another. 5. Suppose that you have two lines L1 and L2 and a third line M. Suppose that M intersects L1 and L2 at two interior angles whose sum is less than the sum of two right angles on one side. Then L1 and L2 meet on that side. (See Figure 1.) L1 M L2 Figure 1: Euclid’s fifth postulate. 1 Euclid’s fifth postulate is often reformulated like this: For any line L and any point p not on L, there is a unique line L′ through p such that L and L′ do not intersect – i.e., are parallel. This is why Euclid’s fifth postulate is often called the parallel postulate. Euclid’s postulates have a quaint and vague sound to modern ears. They have a number of problems. In particular, they do not uniquely pin down the Euclidean plane. A modern approach is to make a concrete model for the Euclidean plane, and then observe that Euclid’s postulates hold in that model. So, from a modern point of view, the Euclidean plane is R2, namely the set of ordered pairs of real numbers.
[Show full text]