# Exercise Set 3. Similar triangles. Parallel lines

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 interior point of the side BC. The perpendicular from the point D to the line BC intersects the lines AB and AC in the points E and F. Show that the sum DE + DF remains constant as D moves along BC. (2) Let ABC be a triangle with AB = AC. Let D be an interior point of the side BC. The perpendicular from the point D to the line AB intersects AB at the point E and the perpendicular from D to AC intersects AC at the point F. Show that the sum DE + DF remains constant as D moves along BC. (3) Let ABC be a triangle. Let D be an interior point of the side AB, and E of AC such that BD = CE. Let M and N be the midpoints of the segments DE and BC, respectively. Prove that the line M N is parallel to the bisector of the angle BAC. [Hint: Draw MB parallel and equal to BD, and MC parallel and equal to CE. We call that translating BD and CE so they pass through M.] (4) Let ABCD be a convex quadrilateral and let M, N, P, and Q be interior points of the sides AB, BC, CD and DA respectively, such that the following relations hold: AM MB = DP P C = a and BN NC = AQ QD = b. Let O be the point of intersection of the lines MP and NQ. Show that the following relations hold: QO = a and MO OP = b. ON [Translate AB and CD so that they pass through Q. ] (5) Let ABC and AB C be two triangles such that AB AB = AC AC = BC B C and such that AC lies in the interior of the angle BAB, while AB lies in the interior of the angle ĈAC. Prove that a) the angle between the lines BC and B C is equal to that between AB and AB. b) If M is an interior point of BB and N is an interior point of CC such that MB MB = NC NC show that the triangle AMN is similar to ABC. (6) Let ABC and A B C be two triangles such that BAC = B A C, ÂCB = A C B, ĈBA = C B A (here the angles are measured clockwise, such that, for example, BAC = ĈAB). Let A, B and C be interior points of the segments AA, BB and CC respectively, such that AA AA = BB BB = CC CC. 1

2 2 Show that the triangle A B C is similar to the triangle ABC. [Translate ABC and A B C so that they all pass through A. ] (7) Use Menelaus theorem to prove that the centroid of a triangle divides each median in the ratio [2 : 1]. (8) Let BMCP be a trapezoid with BM CP, and let A be the point of intersection of the lines BP and MC. Let N be the point of intersection of lines MP and CB. Knowing that AM = 3 cm and AC = 7 cm, calculate CN NB. Let B be a point on the line BM and P a point on the line P C such that A, B and P are collinear. Let N be the point of intersection of lines MP and CB. Prove that NN is parallel to BB. (9) (Ceva s Theorem) Consider ABC and P a point in its interior. Let A 1, B 1, C 1 be the intersections of P A with BC, P B with CA, P C with AB, respectively. a) Use Menelaus Theorem in ABA 1 and ACA 1 to prove Ceva s formula: AC 1 C 1 B BA 1 A 1 C CB 1 B 1 A = 1 b) Reciprocally, if the formula above holds for three random points A 1, B 1, C 1 on the three sides of the triangle, prove that the lines AA 1, BB 1 and CC 1 intersect at the same point P. (10) Consider a quadrilateral ABCD. Let M denote the intersection point of lines AB and CD, and let N denote the intersection point of lines BC and AD. Let P be the midpoint of the diagonal AC, let Q be the midpoint of diagonal BD, let R be the midpoint of MN, let X be the midpoint of DN, let Y be the midpoint of CN and Z the midpoint of CD. a) Prove that the points X, Z and Q are collinear. Also prove that X, Y and R are collinear and that Y, Z and P are collinear. b) Use Menelaus Theorem in XY Z to prove that the points P, Q and R are collinear. (11) Let ABC be a triangle and P a point in its interior, not lying on any of the medians of ABC. Let A 1, B 1, C 1 be the intersections of P A with BC, P B with CA, P C with AB, respectively, and let A 2, B 2, C 2 be the intersections of B 1 C 1 with BC, C 1 A 1 with CA, A 1 B 1 with AB, respectively. a) Prove that BA 1 A 1 C = BA 2 A 2 C. Find similar relations on the lines AB and AC. b) Use a) above to prove that the points A 2, B 2, C 2 are collinear. c) Prove that the midpoints of A 1 A 2, B 1 B 2, C 1 C 2 are collinear.

4 4 (11) (IMO 2008) An acute angled triangle has orthocentre H. The circle passing through H with centre the midpoint of BC intersects the line BC at A 1 and A 2. The circle passing through H with centre the midpoint of AC intersects AC at B 1 and B 2, and the circle passing through H with centre the midpoint of AB intersects AB at C 1 and C 2. Show that A 1, A 2, B 1, B 2, C 1, C 2 lie on a circle. (12) Let ABC be a triangle, let I be the incentre, let I A, I B and I C be the centers of the circles externally tangent to sides. Let A, B, C be the feet of the perpendiculars from I to the sides of the triangle, and A, B, C the feet of the perpendiculars from I A to BC, from I B to AC and from I C to AB, respectively. a) Show that I A A, IBB, I C C are the altitudes of I A I B I C. b) Show that I A A, I B B, I C C intersect at the circumcircle O of I A I B I C, of radius O I A = 2R, where R is the circumradius of ABC. (Hint: use 7) c) Find all pairs of similar triangles in the figure. In particular, prove that A B = B C = C A = I A I B I B I C I C I A r 2R, where r is the inradius of ABC. d) Show that I A A, I B B, I C C intersect at a point S. Furthermore, S is collinear with I, O, the orthocentre of A B C and the centroids of I A I B I B and A B C. e) Let M, N, P be the midpoints of the sides BC, CA and AB respectively. The perpendicular from M to I B I C, from N to I C I A and from P to I A I B intersect at a point Q, such that Q, I and G are collinear, where G is the centroid of ABC, and IG = 2QG. (13) (Ptolemy s relation) Show that ABCD is a cyclic quadrilateral if and only if AB CD + AD BC = AC BD. (14) Let ABC be a triangle with AB = AC = BC and let P be a point on its circumcircle. Show that the sum P A + P B + P C does not depend on the position of P on the circumcircle. (15) (IMO 2007) In triangle ABC the bisector of the angle BCA intersects the circumcircle again at R, the perpendicular bisector of BC at P, and the perpendicular bisector of AC at Q. The midpoint of BC is K and the midpoint of AC is L. Prove that the triangles RP K and RQL have the same area. (16) Let ABCD be a cyclic quadrilateral and let P be the intersection point of its diagonals. Show that there exists a point Q such that the triangles P BC and QBC are similar, and so are triangles P AD and QAD. (17) Let ABC and A B C be similar to each other as in question 6 (such that the orientation of the angles counts, too). Choose points A, B and C such that the triangles AA A, BB B and CC C are all similar to each other, (again, the orientation of the angles counts). Guess what? (Hint: find a centre of similarity, like in question 16). (18) (IMO 2009) Let ABC be a triangle with circumcentre O. The points P and Q are interior points of the sides CA and AB, respectively. Let K, L, and M be the midpoints of the segments BP, CQ and P Q, respectively, and let Γ be the circle passing through K, L, and M. Suppose that the line P Q is tangent to the circle Γ. Prove that OP =OQ.

5 5 Anca Mustata, School of Mathematical Sciences, UCC address:

### 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

### Chapter 3. Inversion and Applications to Ptolemy and Euler

Chapter 3. Inversion and Applications to Ptolemy and Euler 2 Power of a point with respect to a circle Let A be a point and C a circle (Figure 1). If A is outside C and T is a point of contact of a tangent

### Triangle Centers MOP 2007, Black Group

Triangle Centers MOP 2007, Black Group Zachary Abel June 21, 2007 1 A Few Useful Centers 1.1 Symmedian / Lemmoine Point The Symmedian point K is defined as the isogonal conjugate of the centroid G. Problem

### CHAPTER 1. LINES AND PLANES IN SPACE

CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given

### Contents. Problems... 2 Solutions... 6

Contents Problems............................ Solutions............................ 6 Problems Baltic Way 014 Problems Problem 1 Show that cos(56 ) cos( 56 ) cos( 56 )... cos( 3 56 ) = 1 4. Problem Let

### Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

### Section 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.

. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### The Inversion Transformation

The Inversion Transformation A non-linear transformation The transformations of the Euclidean plane that we have studied so far have all had the property that lines have been mapped to lines. Transformations

### 2009 Chicago Area All-Star Math Team Tryouts Solutions

1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total

### Regents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014

Regents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014 i May 2014 777777 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 New York

### GeoGebra. 10 lessons. Gerrit Stols

GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter

### EGMO 2013. Problems with Solutions

EGMO 2013 Problems with Solutions Problem Selection Committee: Charles Leytem (chair, Pierre Haas, Jingran Lin, Christian Reiher, Gerhard Woeginger. The Problem Selection Committee gratefully acknowledges

### 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

### Geo, Chap 4 Practice Test, EV Ver 1

Class: Date: Geo, Chap 4 Practice Test, EV Ver 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (4-3) In each pair of triangles, parts are congruent as

### Acceptance Page 2. Revision History 3. Introduction 14. Control Categories 15. Scope 15. General Requirements 15

Acceptance Page 2 Revision History 3 Introduction 14 Control Categories 15 Scope 15 General Requirements 15 Control Category: 0.0 Information Security Management Program 17 Objective Name: 0.01 Information

### Mean Geometry. Zachary R. Abel. July 9, 2007

Mean Geometry Zachary R. bel July 9, 2007 ontents Introduction 3. asic efinitions............................................... 3.2 oint verages................................................ 3.3 Figure

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### CHAPTER FIVE. 5. Equations of Lines in R 3

118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

### (15.) To find the distance from point A to point B across. a river, a base line AC is extablished. AC is 495 meters

(15.) To find the distance from point A to point B across a river, a base line AC is extablished. AC is 495 meters long. Angles

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### 6 Commutators and the derived series. [x,y] = xyx 1 y 1.

6 Commutators and the derived series Definition. Let G be a group, and let x,y G. The commutator of x and y is [x,y] = xyx 1 y 1. Note that [x,y] = e if and only if xy = yx (since x 1 y 1 = (yx) 1 ). Proposition

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

### DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

### 2006 George Baloglou first draft: summer 2001 CHAPTER 7 COMPOSITIONS OF ISOMETRIES

2006 George Baloglou first draft: summer 2001 CHAPTER 7 COMPOSITIONS OF ISOMETRIES 7.0 Isometry hunting 7.0.1 Nothing totally new. Already in 4.0.4 we saw that the composition (combined effect) of a rotation

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### College of Charleston Math Meet 2008 Written Test Level 1

College of Charleston Math Meet 2008 Written Test Level 1 1. Three equal fractions, such as 3/6=7/14=29/58, use all nine digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly one time. Using all digits exactly one

### David Bressoud Macalester College, St. Paul, MN. NCTM Annual Mee,ng Washington, DC April 23, 2009

David Bressoud Macalester College, St. Paul, MN These slides are available at www.macalester.edu/~bressoud/talks NCTM Annual Mee,ng Washington, DC April 23, 2009 The task of the educator is to make the

### Factoring, Solving. Equations, and Problem Solving REVISED PAGES

05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

### Star and convex regular polyhedra by Origami.

Star and convex regular polyhedra by Origami. Build polyhedra by Origami.] Marcel Morales Alice Morales 2009 E D I T I O N M O R A L E S Polyhedron by Origami I) Table of convex regular Polyhedra... 4

### JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

### SECTION 1-6 Quadratic Equations and Applications

58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be

### 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

### Trigonometry WORKSHEETS

WORKSHEETS The worksheets available in this unit DO NOT constitute a course since no instructions or worked examples are offered, and there are far too many of them. They are offered here in the belief

### Online EFFECTIVE AS OF JANUARY 2013

2013 A and C Session Start Dates (A-B Quarter Sequence*) 2013 B and D Session Start Dates (B-A Quarter Sequence*) Quarter 5 2012 1205A&C Begins November 5, 2012 1205A Ends December 9, 2012 Session Break

### Indiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A.

STUDENT S SOLUTIONS MANUAL JUDITH A. PENNA Indiana Universit Purdue Universit Indianapolis COLLEGE ALGEBRA: GRAPHS AND MODELS FIFTH EDITION Marvin L. Bittinger Indiana Universit Purdue Universit Indianapolis

### 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,

### http://school-maths.com Gerrit Stols

For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It

### Fundamentals of Geometry. Oleg A. Belyaev belyaev@polly.phys.msu.ru

Fundamentals of Geometry Oleg A. Belyaev belyaev@polly.phys.msu.ru February 28, 2007 Contents I Classical Geometry 1 1 Absolute (Neutral) Geometry 3 1.1 Incidence....................................................

### Victims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years

Claim#:021914-174 Initials: J.T. Last4SSN: 6996 DOB: 5/3/1970 Crime Date: 4/30/2013 Status: Claim is currently under review. Decision expected within 7 days Claim#:041715-334 Initials: M.S. Last4SSN: 2957

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### CROSS REFERENCE. Cross Reference Index 110-122. Cast ID Number 110-111 Connector ID Number 111 Engine ID Number 112-122. 2015 Ford Motor Company 109

CROSS REFERENCE Cross Reference Index 110-122 Cast ID Number 110-111 Connector ID Number 111 112-122 2015 Ford Motor Company 109 CROSS REFERENCE Cast ID Number Cast ID Ford Service # MC Part # Part Type

### Simple trigonometric substitutions with broad results

Simple trigonometric substitutions with broad results Vardan Verdiyan, Daniel Campos Salas Often, the key to solve some intricate algebraic inequality is to simplify it by employing a trigonometric substitution.

### Support Materials for Core Content for Assessment. Mathematics

Support Materials for Core Content for Assessment Version 4.1 Mathematics August 2007 Kentucky Department of Education Introduction to Depth of Knowledge (DOK) - Based on Norman Webb s Model (Karin Hess,

### 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

### Unit 5 Area. What Is Area?

Trainer/Instructor Notes: Area What Is Area? Unit 5 Area What Is Area? Overview: Objective: Participants determine the area of a rectangle by counting the number of square units needed to cover the region.

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### BALANCED THREE-PHASE CIRCUITS

BALANCED THREE-PHASE CIRCUITS The voltages in the three-phase power system are produced by a synchronous generator (Chapter 6). In a balanced system, each of the three instantaneous voltages have equal

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### 4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

### BX in ( u, v) basis in two ways. On the one hand, AN = u+

1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

### SOL Warm-Up Graphing Calculator Active

A.2a (a) Using laws of exponents to simplify monomial expressions and ratios of monomial expressions 1. Which expression is equivalent to (5x 2 )(4x 5 )? A 9x 7 B 9x 10 C 20x 7 D 20x 10 2. Which expression

### ( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make

### Secondary Mathematics Syllabuses

Secondary Mathematics Syllabuses Copyright 006 Curriculum Planning and Development Division. This publication is not for sale. All rights reserved. No part of this publication may be reproduced without

### Middle School Mathematics

The Praxis Study Companion Middle School Mathematics 5169 www.ets.org/praxis Welcome to the Praxis Study Companion Welcome to The Praxis Study Companion Prepare to Show What You Know You have been working

### 1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

### Topics in Elementary Geometry. Second Edition

Topics in Elementary Geometry Second Edition O. Bottema (deceased) Topics in Elementary Geometry Second Edition With a Foreword by Robin Hartshorne Translated from the Dutch by Reinie Erné 123 O. Bottema

### Identifying second degree equations

Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

### The Handshake Problem

The Handshake Problem Tamisha is in a Geometry class with 5 students. On the first day of class her teacher asks everyone to shake hands and introduce themselves to each other. Tamisha wants to know how

### BALANCED THREE-PHASE AC CIRCUIT

BAANCED THREE-PHASE AC CRCUT Balanced Three-Phase oltage Sources Delta Connection Star Connection Balanced 3-hase oad Delta Connection Star Connection Power in a Balanced Phase Circuit ntroduction Three

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### AUTOMATED PROOF OF GEOMETRY THEOREMS INVOLVING ORDER RELATION IN THE FRAME OF THE THEOREMA PROJECT

KNOWLEDGE ENGINEERING: PRINCIPLES AND TECHNIQUES Proceedings of the International Conference on Knowledge Engineering, Principles and Techniques, KEPT2007 Cluj-Napoca (Romania), June 6 8, 2007, pp. 307

### Appendix A Designing High School Mathematics Courses Based on the Common Core Standards

Overview: The Common Core State Standards (CCSS) for Mathematics are organized by grade level in Grades K 8. At the high school level, the standards are organized by strand, showing a logical progression

### DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2

DOE-HDBK-1014/2-92 JUNE 1992 DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2 U.S. Department of Energy Washington, D.C. 20585 FSC-6910 Distribution Statement A. Approved for public release; distribution

### Lines and Planes 1. x(t) = at + b y(t) = ct + d

1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Graphing and Solving Nonlinear Inequalities

APPENDIX LESSON 1 Graphing and Solving Nonlinear Inequalities New Concepts A quadratic inequality in two variables can be written in four different forms y < a + b + c y a + b + c y > a + b + c y a + b

### Calculation of Valu-Trac Statuses

Calculation of Intrinsic Value Yield Latest Cash Earnings (Net Income + Depreciation and Amortization) (put aside) Dividend (subtract) Provision for Depreciation (Net Assets x Inflation Rate) (subtract)

### "HIGHER EDUCATION VALUES AND OPINIONS SURVEY" ADVANCED PLACEMENT TEACHERS and GUIDANCE COUNSELORS May-June 1994

"HIGHER EDUCATION VALUES AND OPINIONS SURVEY" ADVANCED PLACEMENT TEACHERS and GUIDANCE COUNSELORS May-June 1994 VARIABLE SURVEY ANSWER NAME QUESTION CATEGORIES Facsimile the original mail questionnaire

### PROOFS BY DESCENT KEITH CONRAD

PROOFS BY DESCENT KEITH CONRAD As ordinary methods, such as are found in the books, are inadequate to proving such difficult propositions, I discovered at last a most singular method... that I called the

### Attachment "A" - List of HP Inkjet Printers

HP Deskjet 350c Printer HP Deskjet 350cbi Printer HP Deskjet 350cbi Printer w/roller-case HP Deskjet 420 Printer HP Deskjet 420c Printer HP Deskjet 610c Printer HP Deskjet 610cl Printer HP Deskjet 612c

### Factoring Special Polynomials

6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These

CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER A Click here for answers. S Click here for solutions. sin cos s. Show that for all.. Show that y y 6 for all numbers and y such that and. 3. Let a and b be

### 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

### Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### SERVER CERTIFICATES OF THE VETUMA SERVICE

Page 1 Version: 3.4, 19.12.2014 SERVER CERTIFICATES OF THE VETUMA SERVICE 1 (18) Page 2 Version: 3.4, 19.12.2014 Table of Contents 1. Introduction... 3 2. Test Environment... 3 2.1 Vetuma test environment...

### In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).

CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,

### PROBLEM 2.9. sin 75 sin 65. R = 665 lb. sin 75 sin 40

POBLEM 2.9 A telephone cable is clamped at A to the pole AB. Knowing that the tension in the right-hand portion of the cable is T 2 1000 lb, determine b trigonometr (a) the required tension T 1 in the

### FSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers

FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward

### Warm-Up 1. 1. What is the least common multiple of 6, 8 and 10?

Warm-Up 1 1. What is the least common multiple of 6, 8 and 10? 2. A 16-page booklet is made from a stack of four sheets of paper that is folded in half and then joined along the common fold. The 16 pages

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### YOU CAN COUNT ON NUMBER LINES

Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

### Algorithms for Designing Pop-Up Cards

Algorithms for Designing Pop-Up Cards Zachary Abel Erik D. Demaine Martin L. Demaine Sarah Eisenstat Anna Lubiw André Schulz Diane L. Souvaine Giovanni Viglietta Andrew Winslow Abstract We prove that every

### Cissoid. History. From Robert C. Yates Curves and their properties (1952):

History Cissoid Diocles ( 250 100 BC) invented this curve to solve the doubling of the cube problem (also know as the the Delian problem). The name cissoid (ivy-shaped) derives from the shape of the curve.

### College Prep. Geometry Course Syllabus

College Prep. Geometry Course Syllabus Mr. Chris Noll Turner Ashby High School - Room 211 Email: cnoll@rockingham.k12.va.us Website: http://blogs.rockingham.k12.va.us/cnoll/ School Phone: 828-2008 Text:

### MATHEMATICS A A502/01 Unit B (Foundation Tier)

THIS IS A NEW SPECIFICATION F GENERAL CERTIFICATE OF SECONDARY EDUCATION MATHEMATICS A A502/01 Unit B (Foundation Tier) *A533721112* Candidates answer on the question paper. OCR supplied materials: None

### Sample Problems. Practice Problems

Lecture Notes Factoring by the AC-method page 1 Sample Problems 1. Completely factor each of the following. a) 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 c) 162a + 162b 2ax 4 2bx 4 e) 3a 2 5a 2 b) a 2 x 3 b 2 x

### Shapes Bingo. More general matters which apply to the use of this unit are covered on the next page.

Shapes Bingo Shapes Bingo This unit provides the material for practicing some basic shape identification in the context of the well-known game of Bingo. Directions on how to play Bingo are not given here.

### Assumption College Final Examination Content

Final Examination Content EP-M 3/1 Semester 2 Academic Year 2013 Code: MA20212 Subject: Mathematical Skills 6 books / Pages Sum Final Coordinate Geometry 1. - Cartesian Coordinate System - Rectangular

### SECTION A-3 Polynomials: Factoring

A-3 Polynomials: Factoring A-23 thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume 4

### Chapter 7. Cartesian Vectors. By the end of this chapter, you will

Chapter 7 Cartesian Vectors Simple vector quantities can be expressed geometrically. However, as the applications become more complex, or involve a third dimension, you will need to be able to express

### Number Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures

Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it

### Natural Disaster Recovery and Quadrilaterals

Natural Disaster Recovery and Quadrilaterals I. UNIT OVERVIEW & PURPOSE: In this unit, students will apply their knowledge of quadrilaterals to solve mathematics problems concerning a tornado that struck

### 12.5 Equations of Lines and Planes

Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P