Using a Hands-On Activities Based Approach in Geometry
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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. -
Investigating Centers of Triangles: the Fermat Point
Investigating Centers of Triangles: The Fermat Point A thesis submitted to the Miami University Honors Program in partial fulfillment of the requirements for University Honors with Distinction by Katherine Elizabeth Strauss May 2011 Oxford, Ohio ABSTRACT INVESTIGATING CENTERS OF TRIANGLES: THE FERMAT POINT By Katherine Elizabeth Strauss Somewhere along their journey through their math classes, many students develop a fear of mathematics. They begin to view their math courses as the study of tricks and often seemingly unsolvable puzzles. There is a demand for teachers to make mathematics more useful and believable by providing their students with problems applicable to life outside of the classroom with the intention of building upon the mathematics content taught in the classroom. This paper discusses how to integrate one specific problem, involving the Fermat Point, into a high school geometry curriculum. It also calls educators to integrate interesting and challenging problems into the mathematics classes they teach. In doing so, a teacher may show their students how to apply the mathematics skills taught in the classroom to solve problems that, at first, may not seem directly applicable to mathematics. The purpose of this paper is to inspire other educators to pursue similar problems and investigations in the classroom in order to help students view mathematics through a more useful lens. After a discussion of the Fermat Point, this paper takes the reader on a brief tour of other useful centers of a triangle to provide future researchers and educators a starting point in order to create relevant problems for their students. iii iv Acknowledgements First of all, thank you to my advisor, Dr. -
On the Fermat Point of a Triangle Jakob Krarup, Kees Roos
280 NAW 5/18 nr. 4 december 2017 On the Fermat point of a triangle Jakob Krarup, Kees Roos Jakob Krarup Kees Roos Department of Computer Science Faculty of EEMCS University of Copenhagen, Denmark Technical University Delft [email protected] [email protected] Research On the Fermat point of a triangle For a given triangle 9ABC, Pierre de Fermat posed around 1640 the problem of finding a lem has been solved but it is not known by 2 point P minimizing the sum sP of the Euclidean distances from P to the vertices A, B, C. whom. As will be made clear in the sec- Based on geometrical arguments this problem was first solved by Torricelli shortly after, tion ‘Duality of (F) and (D)’, Torricelli’s ap- by Simpson in 1750, and by several others. Steeped in modern optimization techniques, proach already reveals that the problems notably duality, however, Jakob Krarup and Kees Roos show that the problem admits a (F) and (D) are dual to each other. straightforward solution. Using Simpson’s construction they furthermore derive a formula Our first aim is to show that application expressing sP in terms of the given triangle. This formula appears to reveal a simple re- of duality in conic optimization yields the lationship between the area of 9ABC and the areas of the two equilateral triangles that solution of (F), straightforwardly. Devel- occur in the so-called Napoleon’s Theorem. oped in the 1990s, the field of conic optimi- zation (CO) is a generalization of the more We deal with two problems in planar geo- Problem (D) was posed in 1755 by well-known field of linear optimization (LO). -
Chapter 1 – Symmetry of Molecules – P. 1
Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules. -
Mathematical Gems
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 1 Mathematical Gems ROSS HONSBERGER MATHEMATICAL GEMS FROM ELEMENTARY COMBINATORICS, NUMBER THEORY, AND GEOMETRY By ROSS HONSBERGER THE DOLCIANI MATHEMATICAL EXPOSITIONS Published by THE MArrHEMATICAL ASSOCIATION OF AMERICA Committee on Publications EDWIN F. BECKENBACH, Chairman 10.1090/dol/001 The Dolciani Mathematical Expositions NUMBER ONE MATHEMATICAL GEMS FROM ELEMENTARY COMBINATORICS, NUMBER THEORY, AND GEOMETRY By ROSS HONSBERGER University of Waterloo Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA © 1978 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 73-89661 Complete Set ISBN 0-88385-300-0 Vol. 1 ISBN 0-88385-301-9 Printed in the United States of Arnerica Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 FOREWORD The DOLCIANI MATHEMATICAL EXPOSITIONS serIes of the Mathematical Association of America came into being through a fortuitous conjunction of circumstances. Professor-Mary P. Dolciani, of Hunter College of the City Uni versity of New York, herself an exceptionally talented and en thusiastic teacher and writer, had been contemplating ways of furthering the ideal of excellence in mathematical exposition. At the same time, the Association had come into possession of the manuscript for the present volume, a collection of essays which seemed not to fit precisely into any of the existing Associa tion series, and yet which obviously merited publication because of its interesting content and lucid expository style. It was only natural, then, that Professor Dolciani should elect to implement her objective by establishing a revolving fund to initiate this series of MATHEMATICAL EXPOSITIONS. -
20. Geometry of the Circle (SC)
20. GEOMETRY OF THE CIRCLE PARTS OF THE CIRCLE Segments When we speak of a circle we may be referring to the plane figure itself or the boundary of the shape, called the circumference. In solving problems involving the circle, we must be familiar with several theorems. In order to understand these theorems, we review the names given to parts of a circle. Diameter and chord The region that is encompassed between an arc and a chord is called a segment. The region between the chord and the minor arc is called the minor segment. The region between the chord and the major arc is called the major segment. If the chord is a diameter, then both segments are equal and are called semi-circles. The straight line joining any two points on the circle is called a chord. Sectors A diameter is a chord that passes through the center of the circle. It is, therefore, the longest possible chord of a circle. In the diagram, O is the center of the circle, AB is a diameter and PQ is also a chord. Arcs The region that is enclosed by any two radii and an arc is called a sector. If the region is bounded by the two radii and a minor arc, then it is called the minor sector. www.faspassmaths.comIf the region is bounded by two radii and the major arc, it is called the major sector. An arc of a circle is the part of the circumference of the circle that is cut off by a chord. -
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 -
Symmetry in Chemistry - Group Theory
Symmetry in Chemistry - Group Theory Group Theory is one of the most powerful mathematical tools used in Quantum Chemistry and Spectroscopy. It allows the user to predict, interpret, rationalize, and often simplify complex theory and data. At its heart is the fact that the Set of Operations associated with the Symmetry Elements of a molecule constitute a mathematical set called a Group. This allows the application of the mathematical theorems associated with such groups to the Symmetry Operations. All Symmetry Operations associated with isolated molecules can be characterized as Rotations: k (a) Proper Rotations: Cn ; k = 1,......, n k When k = n, Cn = E, the Identity Operation n indicates a rotation of 360/n where n = 1,.... k (b) Improper Rotations: Sn , k = 1,....., n k When k = 1, n = 1 Sn = , Reflection Operation k When k = 1, n = 2 Sn = i , Inversion Operation In general practice we distinguish Five types of operation: (i) E, Identity Operation k (ii) Cn , Proper Rotation about an axis (iii) , Reflection through a plane (iv) i, Inversion through a center k (v) Sn , Rotation about an an axis followed by reflection through a plane perpendicular to that axis. Each of these Symmetry Operations is associated with a Symmetry Element which is a point, a line, or a plane about which the operation is performed such that the molecule's orientation and position before and after the operation are indistinguishable. The Symmetry Elements associated with a molecule are: (i) A Proper Axis of Rotation: Cn where n = 1,.... This implies n-fold rotational symmetry about the axis. -
CHAPTER 3 Parallel and Perpendicular Lines Chapter Outline
www.ck12.org CHAPTER 3 Parallel and Perpendicular Lines Chapter Outline 3.1 LINES AND ANGLES 3.2 PROPERTIES OF PARALLEL LINES 3.3 PROVING LINES PARALLEL 3.4 PROPERTIES OF PERPENDICULAR LINES 3.5 PARALLEL AND PERPENDICULAR LINES IN THE COORDINATE PLANE 3.6 THE DISTANCE FORMULA 3.7 CHAPTER 3REVIEW In this chapter, you will explore the different relationships formed by parallel and perpendicular lines and planes. Different angle relationships will also be explored and what happens to these angles when lines are parallel. You will continue to use proofs, to prove that lines are parallel or perpendicular. There will also be a review of equations of lines and slopes and how we show algebraically that lines are parallel and perpendicular. 114 www.ck12.org Chapter 3. Parallel and Perpendicular Lines 3.1 Lines and Angles Learning Objectives • Identify parallel lines, skew lines, and parallel planes. • Use the Parallel Line Postulate and the Perpendicular Line Postulate. • Identify angles made by transversals. Review Queue 1. What is the equation of a line with slope -2 and passes through the point (0, 3)? 2. What is the equation of the line that passes through (3, 2) and (5, -6). 3. Change 4x − 3y = 12 into slope-intercept form. = 1 = − 4. Are y 3 x and y 3x perpendicular? How do you know? Know What? A partial map of Washington DC is shown. The streets are designed on a grid system, where lettered streets, A through Z run east to west and numbered streets 1st to 30th run north to south. -
11.4 Arcs and Chords
Page 1 of 5 11.4 Arcs and Chords Goal By finding the perpendicular bisectors of two Use properties of chords of chords, an archaeologist can recreate a whole circles. plate from just one piece. This approach relies on Theorem 11.5, and is Key Words shown in Example 2. • congruent arcs p. 602 • perpendicular bisector p. 274 THEOREM 11.4 B Words If a diameter of a circle is perpendicular to a chord, then the diameter bisects the F chord and its arc. E G Symbols If BG&* ∏ FD&* , then DE&* c EF&* and DGs c GFs. D EXAMPLE 1 Find the Length of a Chord In ᭪C the diameter AF&* is perpendicular to BD&* . D Use the diagram to find the length of BD&* . 5 F C Solution E A Because AF&* is a diameter that is perpendicular to BD&* , you can use Theorem 11.4 to conclude that AF&* bisects B BD&* . So, BE ϭ ED ϭ 5. BD ϭ BE ϩ ED Segment Addition Postulate ϭ 5 ϩ 5 Substitute 5 for BE and ED . ϭ 10 Simplify. ANSWER ᮣ The length of BD&* is 10. Find the Length of a Segment 1. Find the length of JM&* . 2. Find the length of SR&* . H P 12 K C S C M N G 15 P J R 608 Chapter 11 Circles Page 2 of 5 THEOREM 11.5 Words If one chord is a perpendicular bisector M of another chord, then the first chord is a diameter. J P K Symbols If JK&* ∏ ML&* and MP&** c PL&* , then JK&* is a diameter. -
Geometry Course Outline Learning Targets Unit 1: Proof, Parallel, And
Geometry Course Outline Learning Targets Unit 1: Proof, Parallel, and Perpendicular Lines 1-1-1 Identify, describe, and name points, lines, line segments, rays and planes using correct notation. 1-1-2 Identify and name angles. 1-2-1 Describe angles and angle pairs. 1-2-2 Identify and name parts of a circle. 2-1-1 Make conjectures by applying inductive reasoning. 2-1-2 Recognize the limits of inductive reasoning. 2-2-1 Use deductive reasoning to prove that a conjecture is true. 2-2-2 Develop geometric and algebraic arguments based on deductive reasoning. 3-1-1 Distinguish between undefined and defined terms. 3-1-2 Use properties to complete algebraic two-column proofs. 3-2-1 Identify the hypothesis and conclusion of a conditional statement. 3-2-2 Give counterexamples for false conditional statements. 3-3-1 Write and determine the truth value of the converse, inverse, and contrapositive of a conditional statement. 3-3-2 Write and interpret biconditional statements. 4-1-1 Apply the Segment Addition Postulate to find lengths of segments. 4-1-2 Use the definition of midpoint to find lengths of segments. 4-2-1 Apply the Angle Addition Postulate to find angle measures. 4-2-2 Use the definition of angle bisector to find angle measures. 5-1-1 Derive the Distance Formula. 5-1-2 Use the Distance Formula to find the distance between two points on the coordinate plane. 5-2-1 Use inductive reasoning to determine the Midpoint Formula. 5-2-2 Use the Midpoint Formula to find the coordinates of the midpoint of a segment on the coordinate plane. -
Chapter 3: Parallel and Perpendicular Lines
Parallel and Perpendicular Lines • Lessons 3-1, 3-2, and 3-5 Identify angle Key Vocabulary relationships that occur with parallel lines and a • parallel lines (p. 126) transversal, and identify and prove lines parallel • transversal (p. 127) from given angle relationships. • slope (p. 139) Use slope to analyze a line • Lessons 3-3 and 3-4 • equidistant (p. 160) and to write its equation. • Lesson 3-6 Find the distance between a point and a line and between two parallel lines. The framework of a wooden roller coaster is composed of millions of feet of intersecting lumber that often form parallel lines and transversals. Roller coaster designers, construction managers, and carpenters must know the relationships of angles created by parallel lines and their transversals to create a safe and stable ride. You will find how measures of angles are used in carpentry and construction in Lesson 3-2. 124 Chapter 3 Parallel and Perpendicular Lines Richard Cummins/CORBIS Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 3. For Lesson 3-1 Naming Segments Name all of the lines that contain the given point. (For review, see Lesson 1-1.) P Q R 1. Q 2. R 3. S 4. T S T For Lessons 3-2 and 3-5 Congruent Angles Name all angles congruent to the given angle. (For review, see Lesson 1-4.) 1 2 8 5. Є2 6. Є5 7 3 4 6 5 7.