1 Rose- Hulman Undergraduate Mathematics Journal Tangent circles in the hyperbolic disk Megan Ternes a Volume 14, No. 1, Spring 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN a Aquinas College
2 Rose-Hulman Undergraduate Mathematics Journal Volume 14, No. 1, Spring 2013 Tangent circles in the hyperbolic disk Megan Ternes Abstract. Constructions of tangent circles in the hyperbolic disk, interpreted in Euclidean geometry, give us examples of four mutually tangent circles. These are shown to satisfy Descartes s Theorem for tangent circles. We also show that the Archimedes twin circles in the hyperbolic arbelos are usually not hyperbolic congruent, even though they are Euclidean congruent. We include a few construction instructions because all items under consideration require surprisingly few steps. Acknowledgements: We would like to thank those who sponsored the Mohler-Thompson Summer Research Grant for making this research experience possible. I would also like to thank Dr. Elizabeth Jensen and Dr. Chad Gunnoe for organizing the free on-campus housing for the summer s student researchers, Dr. Michael McDaniel for including me in this project and giving me this opportunity, the referee who reviewed this paper, and David Rader, editor of the Rose-Hulman Undergraduate Mathematics Journal.
3 PAGE 14 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 1 Introduction In this quick trip to the Poincaré disk model of hyperbolic geometry, we use compass and straightedge to create unusual combinations of tangent circles, circles that touch at a single point, where if you take the tangent line of that point for each circle, they will be the exact same line, and use nifty shortcuts to keep the creation processes brief. We prove the short-cuts and their applications, including a proof that the Archimedes Twin Circles are not congruent in hyperbolic geometry. Our work uses constructions to create our examples and support our proofs. We invite the reader to make her or his own versions using compass and straightedge or dynamic geometry software. 2 Background Among the axioms underlying Euclidean geometry, there are five that get special attention. Other geometries use some of these same axioms, but they may not use all of them. Axioms are statements that are universally accepted as true without requiring a proof. Hyperbolic geometry follows the first four of these axioms, but not the fifth axiom, the Parallel Postulate. One negation of the Parallel Postulate says, Through a point not on a line, there are many lines parallel to the given line. The only way for Euclidean-mode creatures like us to see such lines is to let them appear bent. The geometry for this bent space is called hyperbolic geometry. Henri Poincaré defined a disk model of hyperbolic geometry in which Euclidean axioms, like two points determine a line, are true, along with the negation of the Parallel Postulate mentioned above. The Poincaré disk is made of the Euclidean points inside a unit disk, not including the boundary. These points have no size of their own, they just mark locations within the disk. The hyperbolic lines are diameters of the disk and arcs of circles orthogonal to the boundary of the disk, that is, the angles at the points where the disk and hyperbolic line meet are π /2 radians. The center of the disk is called O. In hyperbolic geometry, the angle sum of a triangle can be anything between 0 and π radians. Hyperbolic distance is not the same as Euclidean distance because the length of a segment depends on its location in the disk relative to the boundary. A small segment near O is much shorter than a segment of the same Euclidean size near the boundary. A hyperbolic circle looks Euclidean, but the position of its hyperbolic center depends on the circle s location. That means any hyperbolic circle whose center is not O has two centers: one Euclidean and one hyperbolic, E and H in Figure 1 respectively.
4 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 PAGE 15 Figure 1: Hyperbolic circle with Euclidean and hyperbolic centers. The first axiom states that two points determine a line. Suppose you have two points A and B inside the disk. When the two points are collinear with the center O, the hyperbolic line is simply the diameter of the disk. When the two points are not collinear with O, it can be confirmed that we really have a hyperbolic line by making sure that the line goes through a point and that point s inverse, which is outside of the hyperbolic space. If A is the point we are starting with, then let the inversion of A across the boundary be called. All hyperbolic lines that pass through A are also going to pass through. Point B can be anywhere else inside the disk. We then have three points and can make a Euclidean circle that will be orthogonal to the disk and will therefore be a hyperbolic line. Drawing the circle simply requires finding the intersection of the perpendicular bisectors of two segments made from our three points. Euclid s second axiom is that lines extend indefinitely. Although this appears not to be the case in hyperbolic geometry, hyperbolic lines only have finite Euclidean length. The hyperbolic distance formula where A and B are points inside the space and C and D are where the hyperbolic line through A and B intersects the boundary is the absolute value of ln( ), with complex coordinates. That circles exist is the third axiom. Hyperbolic circles are Euclidean circles, but their hyperbolic centers are not the same as the Euclidean centers unless the hyperbolic circle is centered at O. The Euclidean center of a hyperbolic circle has no hyperbolic significance, but it is useful in constructing hyperbolic circles or finding hyperbolic centers. The fourth axiom says that all right angles are equal. Angles in hyperbolic geometry are measured by their tangent lines, and of course right angles are always π /2 radians. Euclid s fifth axiom, the Parallel Postulate, states that through a point not on a line, there is exactly one line through that point which is parallel to the original line. Due to the curvature of hyperbolic space though, there are many hyperbolic lines that can pass through a point not on a given line which are parallel to that line. Figure 2 depicts a few hyperbolic lines through a point P not on the given red hyperbolic line, which never touch the red hyperbolic line and are therefore parallel to that hyperbolic line.
5 PAGE 16 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 Figure 2: Hyperbolic parallel lines. Constructions in the hyperbolic disk employ inversion of points across the boundary circle. For a point P in the disk, there exists a point on ray such that. We will not need the construction of inversion [Goodman-Strauss], but we do need its properties: any circle through a point and its inverse is orthogonal to the boundary circle. This is a handy way to make sure a circle we construct actually contains a hyperbolic line. We may use points like on ray outside the disk because we are using Euclidean construction tools to make our hyperbolic objects. 3 Our Hyperbolic Toolbox There are some excellent tricks for constructing hyperbolic objects and geometers have probably known many moves without publishing them [Goodman-Strauss]. Certainly constructions for the hyperbolic midpoint, angle bisector, and other basics have been known quite a while [Ross], [McDaniel]. Here are construction steps for a few structures that are quick and easy to do, and we prove why these shortcut steps work. Let s start by making a simple hyperbolic line so we have something to work with. All we need to build a hyperbolic line is two tangent lines. Specifically, for two points A and B on the boundary, their tangents meet at the center, C, of a circle orthogonal to circle O, through A and B.
6 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 PAGE 17 Figure 3: Simple hyperbolic line construction. Proposition 1: Given hyperbolic segment, construct the hyperbolic midpoint M by first constructing E, the intersection of the tangents at A and B. The intersection of and is M. Figure 4: The hyperbolic midpoint construction. Proof: The ray is orthogonal to circle E since E is the Euclidean center. Then the ray contains the hyperbolic center of circle E. The segment is also a diameter of circle E since it is part of the given hyperbolic line and we built circle E orthogonal to this line. Therefore, their intersection, M, is the hyperbolic center of circle E. This makes and hyperbolic radii of the same circle, having the same hyperbolic length. Proposition 2: Given a hyperbolic isosceles triangle with vertex O, the incircle may be constructed as follows. The Euclidean line perpendicular to through D, the Euclidean center of the base of the isosceles triangle, intersects the Euclidean circle containing the base at the point F on the other side of D from ray. The point C is the Euclidean and hyperbolic midpoint of the base, easily found by drawing the Euclidean segment, intersecting at C. The Euclidean ray intersects at Y, a point of tangency for the desired incircle. The Euclidean line perpendicular to at Y intersects at the incenter I.
7 PAGE 18 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 Figure 5: A point of tangency. Proof: Given an isosceles hyperbolic triangle with a vertex at O, it can be proved that line is perpendicular to hyperbolic segment, and thus that Y is the point of tangency on of the incircle. The ray is the angle bisector of AOB, so the Euclidean incenter lies somewhere along that ray between O and C. Point F is the center of the hyperbolic angle bisector of OBA. To see that this is so, we modify Figure 5 and draw the perpendiculars which go with the tangents in question. Figure 6: Point F is center of hyperbolic angle bisector. The following angle sums are all 90 degrees: LBF + FBN = KBL + LBF and OBK + FBN + NBM = NFB + FBN + NBM. The Euclidean isosceles triangle FBN gives us NFB = FBN. These equations imply OBK = KBL. The remainder of the proof uses Figure 5.
8 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 PAGE 19 Thus, C is the intersection of hyperbolic line and AOB s angle bisector. Point Y is the intersection of Euclidean line and hyperbolic segment, so points F, C, and Y are collinear. Segments and are radii of circle D and are therefore the same Euclidean length. Segment is then the third side of isosceles triangle CDF, meaning FCD and CFD are equal. Lines and intersect at point C, making FCD and YCI vertical angles. Since Y and C are both points on the circle with center I, and are Euclidean radii and the same Euclidean length. Segment is then the third side of a second isosceles triangle, and since YCI was shown to be the same as FCD and CFD, then CYI must also be the same size. The angles formed by lines,, and show that the lines are an example of alternate interior angles, and that is therefore parallel to line. Line must be perpendicular to then, because line is the perpendicular bisector of which is on line. 4 Descartes s Theorem in the Hyperbolic Disk In this section we find mutually tangent circles where three of the circles are also hyperbolic lines. Our first excursion into the tangent circles which we find in hyperbolic geometry will be an example of Descartes s Theorem for mutually tangent circles [Wells]. The theorem says if four circles are mutually tangent then the curvatures k 1, k 2, k 3, and k 4 satisfy the equation Any reader handy with constructions is urged to construct four mutually tangent circles: some planning is required. All the construction marks for our example are in the figure below: a remarkably economic use of a few circles and lines in order to get four circles, each one of which is tangent to the other three. These are circles centered at A, B, G and C. (There are other ways to arrange four mutually tangent circles besides one circle containing the other three.) (1)
9 PAGE 20 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 Figure 7: Four tangent circles A, C, G, B. The construction starts with circle O and a pair of perpendicular radii. Finishing the square gives us G, the center of a circle orthogonal to circle O and with the same size Euclidean radius as circle O. The position of point A can be anywhere along that side of the square and the construction holds. The choice of A decides the tangent at T used to find point B, giving two hyperbolic lines, circles A and B. Proposition 3: For this arrangement of hyperbolic lines in a quadrant of the hyperbolic disk, the radical axis of circle O and the hyperbolic line the same Euclidean size as circle O intersects these two interior hyperbolic lines at the points of tangency of a fourth circle, which is also tangent to the first hyperbolic line. Proof. Let s assign the radii a and b to circles A and B, respectively. We will set the radius of the hyperbolic disk, which has the same radius of the first hyperbolic line, to 1. The Pythagorean Theorem applied to the right triangle ABC gives us (2) Equation (2) simplifies to (3) Our construction uses the meeting of the Euclidean lines and to give the center C of a circle tangent to circles A and B. The dimensions of rectangle AGBC show the radius of the
10 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 PAGE 21 fourth circle is also! We built circle C tangent to circles A and B; we will now verify it is tangent to the outer circle. We can show that circle C contains the intersection of the line and circle G. The equations ( ) and combined with give us and. All we have to do now is establish this ordered pair is ab from point C. The square of the distance is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) There is another circle we can construct tangent to three hyperbolic lines in this figure, with very little work. Since any diameter of the disk is a hyperbolic line, we could send two more hyperbolic lines through O which intersect the known points of tangency J and K. This allows us to apply our toolbox shortcut for the incircle of an isosceles hyperbolic triangle with a vertex at O. Our new circle has center W in Figure 8. Since inversion preserves incidence, we know circle W is tangent to the circles G, B and A. Knowing they are mutually tangent means we can use Descartes s Theorem to calculate the radius of circle W. Recall the theorem says that the curvatures k 1, k 2, k 3, and k 4 of four mutually tangent circles satisfy the equation (1). There is one more little hassle: when one of the mutual tangent circles contains the other three, its curvature takes a negative sign in the equation. Some authors call this signed curvature the bend.
11 PAGE 22 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 Figure 8: Another example of Descartes s theorem. The radius of circle W turns out to be ; the interested reader may find this value by letting the unknown radius be x in Descartes s theorem and rewriting the theorem as a quadratic, (the other root of the quadratic is the radius of the first circle, ab, because we are using the same three circles to find the fourth.) So, starting with circles A, B, and G, Descartes s theorem allows two possible sizes and we found a circle of each size. We can apply Proposition 2 to this same drawing to place another tangent circle in this same figure. Our construction steps are all in Euclidean terminology. Reflecting across creates the isosceles triangle OPN. The newer marks in purple show the steps to the incircle of triangle OPN: perpendicular to to find F, intersects at the point of tangency M, and the perpendicular to at M finds the Euclidean center L of our desired circle. Figure 9 contains this new tangent circle.
12 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 PAGE 23 Figure 9: Application of Proposition 2. Geometers might recognize the string of circles in Figure 8 as the start of a Pappus chain. A Pappus chain is a set of tangent circles arranged inside one big circle around one inner, small circle, so that each member of the chain is tangent to its neighbors [Boas]. The Pappus chain is meant to be inside an arbelos, the topic of our next section. 5 The Hyperbolic Arbelos Our last source of tangent circles is the hyperbolic arbelos. Here we will look at the hyperbolic version of Archimedes twin circles, but before we look at the arbelos in the disk, let s recall the much more famous Euclidean version. The boundary of a Euclidean arbelos is formed by three semicircles. The diameters of each circle all lie along the same line. The ends of each circle s diameter are the points of tangency for one of the other two circles. The arbelos has many properties, but the one we are concerned with is the twin circles. For the twin circles, there is a line perpendicular to the semicircles diameters at the point where the two smaller semicircles are tangent to each other. One twin circle is on either side of this line, tangent to it as well as the semicircle above and below it. It has been proved that these two circles are congruent.
13 PAGE 24 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 Euclidean arbelos Archimedes twin circles Figure 10: The arbelos and the twin circles. The arbelos popped up on mathematicians radar recently when Harold Boas published an article on the arbelos in the American Mathematical Monthly, March With references in Greek, Japanese, German, French, and English and a title with a math pun, Reflections on the Arbelos is a truly outstanding read. Boas won the 2009 Lester R. Ford writing award from the MAA for the article [Boas]. The construction of the twin circles pictured in the figure above is not our own construction, so we did not include the construction marks. Each twin is tangent to two of the three circles in the arbelos. They are called twins because they are congruent [Boas]. The hyperbolic version of the arbelos actually results in two different looking arbeloi from the same construction. A hyperbolic line contains the hyperbolic centers and the points of tangency for the hyperbolic circles that make the boundaries of the arbeloi. One arbelos falls on each side of the hyperbolic line. In this case, each of the twin circles is tangent to the hyperbolic line perpendicular to the hyperbolic line containing the diameters. Figure 11 below uses just a few construction lines to create the hyperbolic arbelos on a hyperbolic line which is not a diameter. Points A, B, and C are chosen on the given hyperbolic line and tangent lines are constructed at each point. The intersections of the tangents, D, E, and F, are the Euclidean centers of the circles in the hyperbolic arbelos and the hyperbolic centers are where rays,, and would intersect the hyperbolic line. Someone new to the hyperbolic disk might think the arbelos with the Euclidean centers inside is different from the other arbelos. But remember: the Euclidean centers are merely construction aids. They have no special status as hyperbolic points. The arbelos closest to O is congruent to the other arbelos, just as in the Euclidean case.
14 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 PAGE 25 Figure 11: The hyperbolic arbelos. We will continue thinking about the Euclidean and hyperbolic arbelos figures to prove our last claim. Proposition 4: The Archimedes twin circles in a hyperbolic arbelos are congruent in the hyperbolic metric if and only if the inner circles of the arbelos are congruent in that metric. Proof: Any hyperbolic arbelos may be translated so that the point of intersection between the two inner circles becomes O. Now the Euclidean picture of the arbelos matches the hyperbolic. Moreover, the Euclidean construction of the Archimedes twins works the same, producing the picture of the twins in Figure 10. The Euclidean rays from O through each twin s center must contain their respective hyperbolic centers. This gives us a way to compare the hyperbolic length of their diameters. These diameters are Euclidean congruent. The only way they can be hyperbolic congruent is if these diameters are the same Euclidean distance from O. That only happens when the inner circles of the arbelos are congruent.
15 PAGE 26 RHIT UNDERGRAD. MATH. J., VOL. 14, NO. 1 References Boas, Harold P. Reflections on the Arbelos. The American Mathematical Monthly. 113 (2006): Goodman-Strauss, Chaim. Compass and Straightedge in the Poincaré Disk. The American Mathematical Monthly. 108 (2001): McDaniel, Michael. The Hyperbolic Tourist. Math Horizons (2006): Ross, Skyler. Non-Euclidean Geometry, M.A. Thesis, University of Maine, Wells, David. The Penguin Dictionary of Curious and Interesting Geometry, Penguin, 1992.
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry Put your name here: Score: Instructions: For this lab you will be using the applet, NonEuclid, created by Castellanos, Austin, Darnell,
Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
rchimedes and the rbelos 1 obby Hanson October 17, 2007 The mathematician s patterns, like the painter s or the poet s must be beautiful; the ideas like the colours or the words, must fit together in a
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
IRLES N VOLUME Lesson 1: Introducing ircles ommon ore Georgia Performance Standards M9 12.G..1 M9 12.G..2 Essential Questions 1. Why are all circles similar? 2. What are the relationships among inscribed
12. Parallels Given one rail of a railroad track, is there always a second rail whose (perpendicular) distance from the first rail is exactly the width across the tires of a train, so that the two rails
Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
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
alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
The Euler Line in Hyperbolic Geometry Jeffrey R. Klus Abstract- In Euclidean geometry, the most commonly known system of geometry, a very interesting property has been proven to be common among all triangles.
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
CK-12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.
High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
78 MATHEMATICS INTRODUCTION TO EUCLID S GEOMETRY CHAPTER 5 5.1 Introduction The word geometry comes form the Greek words geo, meaning the earth, and metrein, meaning to measure. Geometry appears to have
The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,
Elements of Plane Geometry by LK These are notes indicating just some bare essentials of plane geometry and some problems to think about. We give a modified version of the axioms for Euclidean Geometry
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M9-1.G..1 Prove that all circles are similar. M9-1.G.. Identify and describe relationships
efinition: circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. We use the symbol to represent a circle. The a line segment from the center
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)
Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides
CONDENSED LESSON 6.1 Tangent Properties In this lesson you will Review terms associated with circles Discover how a tangent to a circle and the radius to the point of tangency are related Make a conjecture
1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,
Canadian Mathematics Competition An activity of The Centre for Education in Ma thematics and Computing, University of W aterloo, Wa terloo, Ontario 2004 Solutions Ga lois Contest (Grade 10) 2004 Waterloo
Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
Unit 3: Circles and Volume This unit investigates the properties of circles and addresses finding the volume of solids. Properties of circles are used to solve problems involving arcs, angles, sectors,
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
dvanced Euclidean Geometry What is the center of a triangle? ut what if the triangle is not equilateral?? Circumcenter Equally far from the vertices? P P Points are on the perpendicular bisector of a line
Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal.
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology firstname.lastname@example.org 1 iameter of incircle T Lemma 1. Let the incircle of triangle
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood
Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This
INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE ABSTRACT:- Vignesh Palani University of Minnesota - Twin cities e-mail address - email@example.com In this brief work, the existing formulae
Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3 The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs
GEOMETRI MENSURTION Question 1 (**) 8 cm 6 cm θ 6 cm O The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 6 cm and the length of the
11 th nnual Harvard-MIT Mathematics Tournament Saturday February 008 Individual Round: Geometry Test 1.  How many different values can take, where,, are distinct vertices of a cube? nswer: 5. In a unit
The Geometry of a Circle Geometry (Grades 10 or 11) A 5 day Unit Plan using Geometers Sketchpad, graphing calculators, and various manipulatives (string, cardboard circles, Mira s, etc.). Dennis Kapatos
Circle s circle is a set of points in a plane that are a given distance from a given point, called the center. The center is often used to name the circle. T This circle shown is described an OT. s always,
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
eometry hapter 10 Study uide Name Terms and Vocabulary: ill in the blank and illustrate. 1. circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.
CONDENSED LESSON 3.1 Duplicating Segments and ngles In this lesson, you Learn what it means to create a geometric construction Duplicate a segment by using a straightedge and a compass and by using patty
tmcs-szilasi 2012/3/1 0:14 page 175 #1 10/1 (2012), 175 181 Classical theorems on hyperbolic triangles from a projective point of view Zoltán Szilasi Abstract. Using the Cayley-Klein model of hyperbolic
Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents Name Chords and Circles: A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
SIMSON S THEOREM MARY RIEGEL Abstract. This paper is a presentation and discussion of several proofs of Simson s Theorem. Simson s Theorem is a statement about a specific type of line as related to a given
Situation: Proving Quadrilaterals in the Coordinate Plane 1 Prepared at the University of Georgia EMAT 6500 Date Last Revised: 07/31/013 Michael Ferra Prompt A teacher in a high school Coordinate Algebra
6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which
MATH STUDENT BOOK 8th Grade Unit 6 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
Krystin Wright Geometer s Sketchpad Assignment Name Date We are going to investigate what happens when we draw the three angle bisectors of a triangle using Geometer s Sketchpad. First, open up Geometer
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
Chapter 3 NON-EUCLIDEN GEOMETRIES In the previous chapter we began by adding Euclid s Fifth Postulate to his five common notions and first four postulates. This produced the familiar geometry of the Euclidean
Basic Geometry Review For Trigonometry Students 16 June 2010 Ventura College Mathematics Department 1 Undefined Geometric Terms Point A Line AB Plane ABC 16 June 2010 Ventura College Mathematics Department