# 13 Groups of Isometries

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 13 Groups of Isometries For any nonempty set X, the set S X of all one-to-one mappings from X onto X is a group under the composition of mappings (Example 71(d)) In particular, if X happens to be the Euclidean plane E, then E is the set of all points in the plane and S E is a group We note that E is not merely an ordinary set of points An important feature of E is that there is a measure of distance between the points of E Among the mappings in S E, we examine those functions which preserve the distance between any two points Clearly, such functions will be more important than other ones in S E, since such mappings respect an important structure of the Euclidean plane E We choose an arbitrary but fixed cartesian coordinate system on E Each point P in E will then be represented by the ordered pair (x,y) of its coordinates We will not distinguish between the point P and the ordered pair (x,y) So we write (x,y) in place of P, where S E The distance between two points P, in E is given by (x 1 x 2 ) 2 +(y 1 y 2 ) 2, if P and have coordinates (x 1,y 1 ),(x 2,y 2 ), respectively This distance will be denoted by d(p,) or by d((x 1,y 1 ),(x 2,y 2 )) 131 Definition: A mapping S E is called an isometry (of E) if for any two points P, in E d(p, ) = d(p,) This word is derived from "isos" and "metron", meaning "equal" and "measure" in Greek The set of all isometries of E will be denoted by Isom E Since the identity mapping E : E E is evidently an isometry, Isom E is a nonempty subset of S E In fact, Isom E S E 132 Theorem: Isom E is a subgroup of S E 119

2 Proof: We must show that the product of two isometries and the inverse of an isometry are isometries (Lemma 92) (i) Let, Isom E Then, for any two points P, in E d(p, ) = d((p ),( ) ) = d(p, ) (since is an isometry) = d(p,) (since is an isometry), so Isom E Hence Isom E is closed under multiplication (ii) Let Isom E and let P, be any two points in E Since S E, there are uniquely determined points P, in E such that P = P, = Thus P = P 1, = 1 Then Hence Isom E S E d(p, ) = d(p, ) (since is an isometry) d(p 1, 1 ) = d(p,) 1 Isom E We examine some special types of isometries, namely translations, rotations and reflections Loosely speaking, a translation shifts every point of E by the same amount in the same direction In more detail, a translation is a mapping which "moves" any point (x,y) in E by a units in the direction of the x-axis and by b units in the direction of the y-axis (the directions being reversed when a or b is negative) See Figure 1 The formal definition is as follows 133 Definition: A mapping E E is called a translation if there are two real numbers a,b such that (x,y) (x + a,y + b) for all points (x,y) in E under this mapping The translation (x,y) (x + a,y + b) will be denoted by a,b 120

3 (x+a,y+b) (x,y) (a,b) (0,0) Figure1 134 Lemma: Let a,b and c,d be arbitrary translations (1) a,b c,d = a+c,b+d (2) 0,0 = E = (3) a, b a,b = = a,b a, b Proof: (1) We have (x,y)( a,b c,d ) = ((x,y) a,b ) c,d for all (x,y) E Thus a,b c,d = a+c,b+d = (x + a,y + b) c,d = ((x + a) + c,(y + b) + d) = (x + (a + c),y + (b + d)) = (x,y) a+c,b+d (2) We have (x,y) 0,0 = (x + 0,y + 0) = (x,y) = (x,y) for all (x,y) E Thus 0,0 = (3) From (1) and (2) we get a, b a,b = ( a)+a,( b)+b = 0,0 a,b a, b = a+( a),b+( b) = 0,0 = = and likewise 135 Lemma: Any translation is an isometry Proof: First of all, we must show that any translation belongs to S E Let a,b be an arbitrary translation (a,b ) There is a mapping : E E such that a,b = = a,b, namely = a, b by Lemma 134(3) Thus a,b is one-to-one and onto by Theorem 317(2) Hence a,b S E Next we show d((x 1,y 1 ),(x 2,y 2 )) = d((x 1,y 1 ) a,b, (x 2,y 2 ) a,b ) for any two points (x 1,y 1 ),(x 2,y 2 ) in E We have 121

4 d((x 1,y 1 ) a,b, (x 2,y 2 ) a,b ) = d((x 1 + a,y 1 + b), (x 2 + a,y 2 + b)) = [(x 1 +a) (x 2 +a)] 2 + [(y 1 +b) (y 2 +b)] 2 = (x 1 x 2 ) 2 +(y 1 y 2 ) 2 = d((x 1,y 1 ),(x 2,y 2 )) and so a,b Isom E 136 Theorem: The set T of all translations is a subgroup of Isom E Proof: Let T = { a,b : a,b } be the set of all translations T is a subset of Isom E by Lemma 135 From Lemma 134(2), = 0,0 T, so T Now we use our subgroup criterion (Lemma 92) a+c,b+d a, b (i) The product of two translations a,b and c,d is a translation T by Lemma 134(1) So T is closed under multiplication (ii) The inverse of any translation a,b T is also a translation T by Lemma 134(3) So T is closed under taking inverses Thus T is a subgroup of Isom E Next we investigate rotations By a rotation about a point C through an angle, we want to understand a mapping from E into E which sends the point C to C and whose effect on any point P C is as follows: we turn the line segment CP about the point C through the angle into a new line segment, say to C; the point P will be sent to the point (see Figure 2) We recall that positive values of measure counterclockwise angles and negative values of measure clockwise angles Rotations are most easily described in a polar coordinate system We choose the center of rotation, the point C, as the pole The initial ray is chosen arbitrarily The point P with polar coordinates (r, ) is then sent to the point whose polar coordinates are (r, + ) If C is the origin and the initial ray is the positive x-axis of a cartesian coordinate system, then the polar and cartesian coordinates of a point P are connected by x = r cos y = r sin 122

5 = (r, + ) P = (r, ) r C Figure 2 In our fixed cartesian coordinate system, the image of any point P = (x,y) can be found as follows If P has polar coordinates (r, ), then its image will have polar coordinates (r, + ), so the cartesian coordinates x,y of := (r, + ) are x = r cos( + ) = r(cos cos sin sin ) = (r cos ) cos (r sin ) sin = x cos y sin, y = r sin( + ) = r(sin cos + cos sin ) = (r sin ) cos + (r cos ) sin = y cos + x sin = x sin + y cos This suggests the following formal definition 137 Definition: A mapping E E is called a rotation about the origin through an angle if there is a real number such that (x,y) (x cos y sin, x sin + y cos ) for all points (x,y) in E under this mapping The rotation (x,y) (x cos y sin, x sin + y cos ) will be denoted by We have an analogue of Lemma 134 for rotations 123

6 138 Lemma: Let and be arbitrary rotations about the origin (1) = (2) 0 = E = + (3) = = Proof: (1) We have (x,y)( ) = ((x,y) ) = (x cos y sin, x sin + y cos ) = ((x cos y sin )cos (x sin +y cos )sin, (x cos y sin )sin + (x sin +y cos )cos ) = (x(cos cos sin sin ) y(sin cos + cos sin ), x(cos sin + sin cos ) + y( sin sin + cos cos )) = (x cos( + ) y sin( + ), x sin( + ) + y cos( + )) = (x,y) + for all (x,y) E Thus = + (2) We have (x,y) 0 = (x cos0 y sin0, x sin0 + y cos0) = (x 0,0 + y) = (x,y) = (x,y) for all (x,y) E Thus 0 = (3) From (1) and (2) we get = ( )+ = 0 = and likewise = +( ) = 0 = Lemma 138 was to be expected When we carry out a rotation through an angle and then a rotation through an angle, we have in effect a rotation through an angle + This is what Lemma 138(1) states Also, when we carry out a rotation through an angle and then a rotation through the same angle in the reverse direction, the final result will be: no net motion at all This is what Lemma 138(3) states 139 Lemma: Any rotation about the origin is an isometry Proof: First of all, we must show that any rotation about the origin belongs to S E Let be an arbitrary rotation about the origin ( ) There is a mapping : E E such that = =, namely = by Lemma 138(3) Thus is one-to-one and onto by Theorem 317(2) Hence S E 124

8 1311 Definition: Let C = (a,b) be a point in E The mapping ( a,b ) 1 a,b : E E is called a rotation about C through an angle (x +a,y +b) (x+a,y+b) (x,y ) (x,y) (a,b) (0,0) Figure 3 We put ( a,b ) 1 R a,b := {( a,b ) 1 a,b : R} This is the set of all rotations about the point (a,b) It is a subgroup of Isom E The proof of this statement is left to the reader Now we examine reflections The cartesian equations of a reflection are very cumbersome For this reason, we give a coordinate-free definition of reflections We need some notation Let P, be distinct points in the plane E In what follows, P will denote the line through P and, and P will denote the line segment between P and So P is the set of points R in E such that d(p,r) + d(r,) = d(p,) The geometric idea of a reflection is that there is a line m and that each point P is mapped to its "mirror image" on the other side of m So P is perpendicular to m and d(p,r) = d(r,), where R is the point of intersection of m and P See Figure 4 m P R Figure 4 126

9 1312 Definition: Let m be a straight line in E and let m : E E be the mapping defined by P m = P if P is on m and P m = if P is not on m and if m is the perpendicular bisector of P m is called the reflection in the line m The perpendicular bisector of P is the line that is perpendicular to P and that intersects P at a point R such that d(p,r) = d(r,) It is also the locus of all points in E which are equidistant from P and So it is the set {R E: d(p,r) = d(,r)} We will make use of this description of the perpendicular bisector in the sequel without explicit mention 1313 Lemma: Let m be the reflection in a line m Then m = m 2 Proof: P m P if P is not on the line m and so m Now we prove that 2 = We have P 2 m m = P( m m ) = (P m ) m = P m = P when P is a point on m by definition It remains to show P 2 m = P also when P is not on m Let P be a point not on m and let = P m, P 1 = m Then is not on m and m is the perpendicular bisector of P as well as of P 1 So P and P 1 are parallel lines Since they have a point in common, they are identical lines Let R be the point at which m and P intersect So P 1 and P 1 is that point on P for which d(,r) = d(p 1,R) Since P is on P and d(,r) = d(p,r), we obtain P = P 1, as was to be proved 1314 Lemma: Any reflection in a line is an isometry Proof: Let m be a line, m the reflection in m and let P, be arbitrary points in the plane We put P 1 = P m, 1 = m We are to show d(p,) = d(p 1, 1 ) We distinguish several cases Case 1 Assume both P and are on m Then P 1 = P and 1 = So d(p,) = d(p 1, 1 ) Case 2 Assume one of the points is on m, the other is not We suppose, without loss of generality, that P is on m and is not on m Let 1 127

10 intersect m at S Then d(,s) = d(s, 1 ), d(p,s) = d(s,p 1 ) since P = P 1 and the angles PS and P 1 1 S are both right angles By the side-angle-side condition, the triangles PS and 1 P 1 S are congruent So the corresponding sides P and P 1 1 have equal length This means d(p,) = d(p 1, 1 ) From now on, assume that neither P nor is on m Let m intersect PP 1 at N and 1 at S P P P m S N S m N T m N S m P=P S T P P P Case 2 Case 3 Case 4 Case 4 Figure 5 Case 3 Assume that P is parallel to m Then the quadrilaterals NPS and NP 1 1 S are rectangles So PP 1 1 is a rectangle and the sides P and P 1 1 have equal length This means d(p,) = d(p 1, 1 ) Case 4 Assume that P is not parallel to m Then P intersects m at a point T As in case 2, PTN and P 1 TN are congruent, and ST and 1 ST are congruent, so d(p,t) = d(p 1,T) and d(,t) = d( 1,T) Also, ST 1 = ST = NTP = NTP 1, which shows that P 1,T, 1 lie on a straight line Then we obtain d(p 1, 1 ) = d(p 1,T) d(t, 1 ) = d(p 1,T) d( 1,T) = d(p,t) d(,t) = d(p,t) d(t,) = d(p,), where the upper or lower sign is to be taken according as whether P, are on the same or on the opposite sides of m 1315 Theorem: Let m be a line in E Then {, E } is a subgroup of Isom m 128

11 Proof: {, } is a finite nonempty subset of Isom E by Lemma 1314 It m is closed under multiplication by Lemma 1313 So it is a subgroup of Isom E by Lemma 93(1) Translations, rotations and reflections are isometries Thus the products of any number of these mappings, carried out in any order, will be isometries, too We show in the rest of this paragraph that all isometries are obtained in this way We need some lemmas 1316 Lemma: Let P,,R be arbitrary points in E (1) There is a unique translation that maps P to (2) If d(p,) = d(p,r), there is a rotation about P that maps to R Proof: (1) When P = (a,b) and = (c,d), say, then m,n maps P to if and only if (a + m,b + n) = (c,d), ie, if and only if m = c a, n = d b So c a,d b is the unique translation that maps P to (2) We draw the circle whose center is at P and whose radius is equal to d(p,) This circle passes through R by hypothesis Let be the angle which the circular arc R subtends at the center P Then a rotation about P through an angle maps to R The next lemma states that an isometry is completely determined by its effect on three points not lying on a line 1317 Lemma: Let P,,R be three distinct points in E that do not lie on a straight line Let, be isometries such that P = P, =, R = R Then = 129

12 P N N Figure 6 Proof: We put 1 = We suppose and try to reach a contradiction If, then there is a point N in E such that N N Since P = P by hypothesis, P = P and so P N Similarly N and R N Now is an isometry, so d(p,n ) = d(p,n ) = d(p,n) and likewise d(,n ) = d(,n) and d(r,n ) = d(r,n) So the circle with center at P and radius d(p,n) and the circle with center at and radius d(,n) intersect at the points N and N Then P is the perpendicular bisector of N N Here we used N N But d(r,n ) = d(r,n) and R lies therefore on the perpendicular bisector of N N, ie, R lies on P, contrary to the hypothesis that P,,R do not lie on a straight line Hence necessarily = and = 1318 Theorem: Let P,,R be three distinct points in E that do not lie on a straight line and let P,,R be three distinct points in E Assume that d(p,) = d(p, ), d(p,r) = d(p,r ), d(,r) = d(,r ) Then there is a translation, a rotation (about an appropriate point and through a suitable angle) and a reflection such that P = P, =, R = R, where denotes or Proof: By Lemma 1316(1), there is a translation that maps P to P We put 1 = and R 1 = R Since is an isometry, d(p,) = d(p, ) = d(p, 1 ), so d(p, 1 ) = d(p, ) By Lemma 1316(2), there is a rotation about P that maps 1 to Let us denote this rotation by Then P = P We put 130

13 R 1 = R 2 Here it may happen that R 2 = R Putting = in this case, we have P = P, =, R = R, as claimed Suppose now R 2 R From d(p,r ) = d(p,r) = d(p,r ) = d(p,r 2 ) and d(,r ) = d(,r) = d(,r ) = d(,r 2 ), we deduce that both P and lie on the perpendicular bisector of R R 2 Denoting the reflection in the line P by, we get P = P, = and R 2 = R Putting = in this case, we have P = P, =, R = R, as claimed The proof is summarized schematically below P P P P 1 R R 1 R 2 R 1319 Theorem: Every isometry can be written as a product of translations, rotations and reflections In fact, if is an arbitrary isometry, then there is a translation, a rotation and a reflection such that = or Proof: Let be an arbitrary isometry Choose any three distinct points P,,R in E not lying on a straight line Then d(p,) = d(p, ), d(p,r) = d(p,r ), d(,r) = d(,r ) So the hypotheses of Theorem 1318 are satisfied with P = P, =, R = R and there is a translation, a rotation and a reflection such that P = P, =, R = R, where = or By Lemma 1317, = Thus = or Exercises 1 Let m be the line in E whose cartesian equation is ax + by + c = 0 Show that the reflection m in the line m is given by (u,v) m = (u 2a a 2 + b 2 (au + bv + c), v 2b a 2 + b 2 (au + bv + c)) 131

14 2 Let m and n be two distinct lines intersecting at a point P Show that m n is a rotation about P Through which angle? 3 Let m and n be parallel lines Show that m n is a translation 4 Prove that every rotation and every translation can be written as a product of two reflections 5 Prove that every isometry can be written as a product of reflections 6 A halfturn given by P = (a,b) about a point P = (a,b) is defined as the mapping (x,y) (2a x, 2b y) for all points (x,y) in E Show that any halfturn is an isometry of order two Prove that the product of three halfturns is a halfturn 7 Prove that a halfturn P is the product of any two reflections in lines intersecting perpendicularly at P 8 Prove that a product of two halfturns is a translation 9 Show that the set of all translations and halfturns is a subgroup of Isom E 10 Prove that the product of four reflections can be written as a product of two reflections 11 Show that 2 /n generates a cyclic subgroup or order n of Isom E 12 Prove that every nonidentity translation is of infinite order and that is of finite order if and only if is a rational multiple of 132

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### What is inversive geometry?

What is inversive geometry? Andrew Krieger July 18, 2013 Throughout, Greek letters (,,...) denote geometric objects like circles or lines; small Roman letters (a, b,... ) denote distances, and large Roman

### Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

### Projective Geometry - Part 2

Projective Geometry - Part 2 Alexander Remorov alexanderrem@gmail.com Review Four collinear points A, B, C, D form a harmonic bundle (A, C; B, D) when CA : DA CB DB = 1. A pencil P (A, B, C, D) is the

### Foundations of Geometry 1: Points, Lines, Segments, Angles

Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Lesson 2: Circles, Chords, Diameters, and Their Relationships

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 measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures

8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

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.

### Circle Name: Radius: Diameter: Chord: Secant:

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

### Projective Geometry. Stoienescu Paul. International Computer High School Of Bucharest,

Projective Geometry Stoienescu Paul International Computer High School Of Bucharest, paul98stoienescu@gmail.com Abstract. In this note, I will present some olympiad problems which can be solved using projective

### DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

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

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations

The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations Dynamic geometry technology should be used to maximize student learning in geometry. Such technology

### Theorem 3.1. If two circles meet at P and Q, then the magnitude of the angles between the circles is the same at P and Q.

3 rthogonal circles Theorem 3.1. If two circles meet at and, then the magnitude of the angles between the circles is the same at and. roof. Referring to the figure on the right, we have A B AB (by SSS),

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

### 3.1 Triangles, Congruence Relations, SAS Hypothesis

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)

### alternate interior angles

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

### GEOMETRY. Constructions OBJECTIVE #: G.CO.12

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

### 1 Solution of Homework

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,

### 4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right

### Bisections and Reflections: A Geometric Investigation

Bisections and Reflections: A Geometric Investigation Carrie Carden & Jessie Penley Berry College Mount Berry, GA 30149 Email: ccarden@berry.edu, jpenley@berry.edu Abstract In this paper we explore a geometric

### SIMSON S THEOREM MARY RIEGEL

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

### www.sakshieducation.com

LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c

### 1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

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

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Triangle Congruence and Similarity A Common-Core-Compatible Approach

Triangle Congruence and Similarity A Common-Core-Compatible Approach The Common Core State Standards for Mathematics (CCSSM) include a fundamental change in the geometry program in grades 8 to 10: geometric

### Three Lemmas in Geometry

Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology yufei.zhao@gmail.com 1 iameter of incircle T Lemma 1. Let the incircle of triangle

### Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

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

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)

Mathematical Sentence - a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement

### 1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

### Angles that are between parallel lines, but on opposite sides of a transversal.

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,

### CIRCLE COORDINATE GEOMETRY

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

### 206 MATHEMATICS CIRCLES

206 MATHEMATICS 10.1 Introduction 10 You have studied in Class IX that a circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). You have

### IMO Geomety Problems. (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition:

IMO Geomety Problems (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points A and B in S, the perpendicular bisector

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Definitions, Postulates and Theorems

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

### Coordinate Plane Project

Coordinate Plane Project C. Sormani, MTTI, Lehman College, CUNY MAT631, Fall 2009, Project XI BACKGROUND: Euclidean Axioms, Half Planes, Unique Perpendicular Lines, Congruent and Similar Triangle Theorems,

### Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

### Elementary triangle geometry

Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors

### Trigonometric Functions and Triangles

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

### 39 Symmetry of Plane Figures

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

### Lesson 19: Equations for Tangent Lines to Circles

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

### Tangents to Circles. by: Joshua Wood. Given two circles, with one completely inside the other, and an arbitrary point on the

Tangents to Circles by: Joshua Wood Given two circles, with one completely inside the other, and an arbitrary point on the larger circle, we constructed script tools in GSP to construct a circle tangent

### Congruence of Triangles

Congruence of Triangles You've probably heard about identical twins, but do you know there's such a thing as mirror image twins? One mirror image twin is right-handed while the other is left-handed. And

### TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

### Chapter 1: Essentials of Geometry

Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

### Incenter Circumcenter

TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

### Lesson 10.1 Skills Practice

Lesson 0. Skills Practice Name_Date Location, Location, Location! Line Relationships Vocabulary Write the term or terms from the box that best complete each statement. intersecting lines perpendicular

### San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS

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

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### Archimedes and the Arbelos 1 Bobby Hanson October 17, 2007

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

### Math 531, Exam 1 Information.

Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)

### Let s Talk About Symmedians!

Let s Talk bout Symmedians! Sammy Luo and osmin Pohoata bstract We will introduce symmedians from scratch and prove an entire collection of interconnected results that characterize them. Symmedians represent

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### Geometry Course Summary Department: Math. Semester 1

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

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

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

### Chapter 6 Notes: Circles

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

### 61. Pascal s Hexagon Theorem.

. Pascal s Hexagon Theorem. Prove that the three points of intersection of the opposite sides of a hexagon inscribed in a conic section lie on a straight line. Hexagon has opposite sides,;, and,. Pascal

### Lesson 18: Looking More Carefully at Parallel Lines

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

Forum Geometricorum Volume 10 (2010) 79 91. FORUM GEOM ISSN 1534-1178 Orthic Quadrilaterals of a Convex Quadrilateral Maria Flavia Mammana, Biagio Micale, and Mario Pennisi Abstract. We introduce the orthic

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

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

### Lesson 19: Equations for Tangent Lines to Circles

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

### C1: Coordinate geometry of straight lines

B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

### Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the

### AREA AND CIRCUMFERENCE OF CIRCLES

AREA AND CIRCUMFERENCE OF CIRCLES Having seen how to compute the area of triangles, parallelograms, and polygons, we are now going to consider curved regions. Areas of polygons can be found by first dissecting

### POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

### 9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Exercise Set 3. Similar triangles. Parallel lines

Exercise Set 3. Similar triangles Parallel lines Note: The exercises marked with are more difficult and go beyond the course/examination requirements. (1) Let ABC be a triangle with AB = AC. Let D be an

### Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

### THREE DIMENSIONAL GEOMETRY

Chapter 11 THREE DIMENSIONAL GEOMETRY 111 Overview 1111 Direction cosines of a line are the cosines of the angles made by the line with positive directions of the co-ordinate axes 111 If l, m, n are the

### IMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao.

ircles Yufei Zhao yufeiz@mit.edu 1 Warm up problems 1. Let and be two segments, and let lines and meet at X. Let the circumcircles of X and X meet again at O. Prove that triangles O and O are similar.

### Chapters 6 and 7 Notes: Circles, Locus and Concurrence

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

### CHAPTER 1 CEVA S THEOREM AND MENELAUS S THEOREM

HTR 1 V S THOR N NLUS S THOR The purpose of this chapter is to develop a few results that may be used in later chapters. We will begin with a simple but useful theorem concerning the area ratio of two

Lecture 24: Saccheri Quadrilaterals 24.1 Saccheri Quadrilaterals Definition In a protractor geometry, we call a quadrilateral ABCD a Saccheri quadrilateral, denoted S ABCD, if A and D are right angles

### GEOMETRY COMMON CORE STANDARDS

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,

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### 2D Geometrical Transformations

2D Geometrical Transformations Translation -Moves points to new locations by adding translation amounts to the coordinates of the points T P(,y) P (,y ) = + d, =y + dy or = y P =P + T + d dy -Totranslate

### Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3

Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3 Main ideas: Identify and use perpendicular bisectors and angle bisectors in triangles. Standard: 12.0 A perpendicular bisector of a

### Terminology: When one line intersects each of two given lines, we call that line a transversal.

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.

### Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

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.

### Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS

1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in