Reflection Property of a Hyperbola
|
|
- Charlene Jefferson
- 6 years ago
- Views:
Transcription
1 Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the rnhes of the hperol, nd whih is direted towrd one fol point of the hperol will e refleted towrd the other fol point when it strikes the rnh of the hperol towrd whih it ws direted. Gerr Del Fio Mth Center Metropolitn Stte Universit St. Pul, Minnesot Jul 6, 9 Definition of the Hperol The reder of this pper is presumed to e fmilir with the definition of hperol nd with the derivtion of the eqution nd orresponding grph of the hperol. The following is psule summr of this definition nd its interprettion. A hperol is the lous of points for whih the solute vlue of the differene of the respetive distnes from n point on the hperol to two fied fol points is onstnt. The grph of suh lous of points, tht is, the grph of hperol, hs the following pperne: Refletion Propert of Hperol Pge
2 P Fol Point Fol Point If P is n ritrr point on either rnh of the hperol, the solute vlue of the distne from one fol point to P minus the distne from the other fol point to P is onstnt, regrdless of the point P tht is seleted on either rnh of the hperol. In the emple tht we re depiting in this pper, we ssume without loss of generlit tht the eqution of the hperol is: where > nd > This ssumption results in the fol points -, ) nd, ) eing loted on the -is where: tht is Furthermore, the onstnt solute vlue of the differene of the respetive distnes etween n point P on either rnh of the hperol nd the two respetive fol points is equl to. The following is the grph of the hperol with its smptotes nd with the oordintes of the -interepts nd the fol points identified: Refletion Propert of Hperol Pge
3 , ) -, ), ) -, ), ) Fol Point, -) Fol Point Definition of the Refletion Propert of the Hperol The following digrm depits the definition of the refletion propert of the hperol: λ Tngent Line λ P Fol Point Fol Point Refletion Propert of Hperol Pge 3
4 3 If P is n ritrr point on n rnh of the hperol, n ngle λ is formed the intersetion of the tngent line t P nd the line from one fol point to the point P. And, n ngle λ of equl mesure is formed the intersetion of the tngent line t P nd the line from the other fol point through the point P. Proof of the Refletion Propert of the Hperol In order to provide n nltil proof of the refletion propert of the hperol, we will mke use of the following digrm: β, ) 4 θ θ -, ), ) β In order to prove the refletion propert, we need to demonstrte tht 3 4. But, we know tht 3 euse these re vertil ngles. And, we know tht 4 euse these re vertil ngles. Therefore, our ojetive in the following proof will e to demonstrte tht. We will omplish this proving tht ) ). We egin otining the slope of the tngent line t the point, ) performing impliit differentition of the eqution of the hperol: Refletion Propert of Hperol Pge 4
5 d d d d Therefore, the slope of the tngent line t the point, ) is: β ) We lso will need β ) whih is determined s follows: β ) β ) Net, we otin the slope of the line from the point -, ) to the point, ) : θ ) Net, we otin the slope of the line from the point, ) to the point, ) : θ ) Sine we will need θ ), we mke the following djustment: θ ) θ ) Now, we proeed to otin ) s follows: β θ Refletion Propert of Hperol Pge 5
6 Refletion Propert of Hperol Pge 6 Therefore: θ β θ β θ β This epression for n e simplified lgerill mking use of the ft tht: Therefore: And, we lso will mke use of the ft tht: Proeeding with the lgeri simplifition, we hve the following:
7 Refletion Propert of Hperol Pge 7 In summr: Net, we proeed to otin s follows: θ β Therefore: θ β θ β θ β In summr: Thus, we hve demonstrted tht: nd Tht is, we hve proven tht:
8 Therefore,. This ompletes the nltil proof of the refletion propert of the hperol. Geometri Eplntion of the Refletion Propert of the Hperol In the preeding pper titled Refletion Propert of n Ellipse, the refletion propert of the ellipse ws eplined in geometri terms using the following digrm: P F M D R E There is one ellipse whih hs the points D nd E s its fol points nd whih psses through the point P. Correspondingl, there is one hperol whih hs the points D nd E s its fol points nd whih psses through the point P. If we eliminte the ellipse from the ove digrm nd reple it with the orresponding hperol, we hve the following digrm: Refletion Propert of Hperol Pge 8
9 P F M D R E In the first of these two digrms ove, the line through points P nd M is tngent to the ellipse nd the line through points R nd P is perpendiulr to the tngent line. In the seond of these two digrms, the respetive roles of these two lines re reversed. Tht is, in the seond of these two digrms ove, the line through points R nd P is tngent to the hperol nd the line through points P nd M is perpendiulr to the tngent line. Refer to Appendi A for supplementr detil.) In either of these two digrms ove, the tringle EPF is n isoseles tringle, nd the line segment from point P to point M is the isetor of ngle EPF. We mde use of this disover in the geometri eplntion of the refletion propert of the ellipse nd in the orollr to the refletion propert of the ellipse. Refletion Propert of Hperol Pge 9
10 In the orollr to the refletion propert of the ellipse, we determined tht the following ngles re equl: φ φ P Ф Ф Ф 3 F M D R E Furthermore, in this digrm ove, ngles φ nd φ 3 re equl euse the re vertil ngles. Therefore: φ φ 3 This ompletes the geometri eplntion of the refletion propert of the hperol. Applition of the Refletion Propert of the Hperol The refletion propert of the hperol n e used in vrious pplitions, suh s the design of high-preision optil equipment. The following is n emple of the design of high-preision telesope whih mkes use of oth hperoli surfe nd proli surfe within the interior of the telesope. This devie is known s Cssegrin refleting telesope whose design ws ttriuted to Lurent Cssegrin, 7th entur siene teher in Frne. Refletion Propert of Hperol Pge
11 The following digrm is two-dimensionl ross-setion of Cssegrin telesope, whih oviousl is three-dimensionl ojet. In the design of this telesope, onve proli mirror forms the k of the telesope. An inoming light r strikes the onve surfe of the proli mirror nd is refleted towrd the onve surfe of hperoli mirror: Inoming Light R Proli Mirror Fol Point of Prol nd Hperol Hperoli Mirror Other Fol Point of Hperol The left-most fol point of the hperol is the sme point s the fol point of the prol. When the light r strikes the onve surfe of the hperol, it is redireted towrd the other fol point of the hperol whih resides t the eepiee of the telesope. Referenes Refletion Propert of Hperol Pge
12 Refletion Propert of Hperol Pge Appendi A This ppendi provides supplementr detil for the geometri eplntion of the refletion propert of the hperol. Refer to the ssoited digrms on pges 8 nd 9 of this pper.) The geometri eplntion mde use of n ellipse nd hperol whih shred the sme fol points nd whose respetive tngent lines t ommon point of intersetion were perpendiulr to eh other. Assume tht the eqution of the ellipse is s follows: > > > nd where nd where Assume tht this ellipse psses through the point ),. Tht is: Assume tht the fol points of this ellipse re -, ) nd, ). Tht is: The hperol whih psses through the point ), nd whih hs its fol points t the points -, ) nd, ) hs n eqution of the following form: B nd where where A B A To prove this ssertion, we first demonstrte tht the point ), lies on the hperol whih hs this given eqution ove: B A
13 Seondl, we demonstrte tht A B. This will verif tht the hperol whih hs the given eqution ove hs the points -, ) nd, ) s its fol points: A B Finll, we will demonstrte tht the line whih is tngent to the ellipse t the point, ) is perpendiulr to the line whih is tngent to the hperol t the point, ). We will omplish this demonstrting tht the negtive reiprol of the slope of the line whih is tngent to the ellipse t the point, ) equls the slope of the line whih is tngent to the hperol t the point, ). Using impliit differentition of the eqution of the ellipse, the slope of the line whih is tngent to the ellipse t the point, ) ws previousl determined to e: The negtive reiprol of this slope is: Using impliit differentition of the eqution of the hperol, the slope of the line whih is tngent to the hperol t the point, ) ws previousl determined to e: B A Refletion Propert of Hperol Pge 3
14 Refletion Propert of Hperol Pge 4 But, we re given the following: B nd A Therefore: A B A B This ompletes the demonstrtion tht the line whih is tngent to the ellipse t the point ), is perpendiulr to the line whih is tngent to the hperol t the point ),.
H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More informationApril 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationm A 1 1 A ! and AC 6
REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:
More information8.3 THE HYPERBOLA OBJECTIVES
8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationSIMPLE NONLINEAR GRAPHS
S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationThe Ellipse. is larger than the other.
The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More informationTHREE DIMENSIONAL GEOMETRY
MD THREE DIMENSIONAL GEOMETRY CA CB C Coordintes of point in spe There re infinite numer of points in spe We wnt to identif eh nd ever point of spe with the help of three mutull perpendiulr oordintes es
More informationEllipses. The second type of conic is called an ellipse.
Ellipses The seond type of oni is lled n ellipse. Definition of Ellipse An ellipse is the set of ll points (, y) in plne, the sum of whose distnes from two distint fied points (foi) is onstnt. (, y) d
More informationIntegration. antidifferentiation
9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationLESSON 11: TRIANGLE FORMULAE
. THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More information3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.
3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationGM1 Consolidation Worksheet
Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More information2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180.
SECTION 8-1 11 CHAPTER 8 Setion 8 1. There re n infinite numer of possile tringles, ll similr, with three given ngles whose sum is 180. 4. If two ngles α nd β of tringle re known, the third ngle n e found
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationfor all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx
Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion
More informationComparing the Pre-image and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationNORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y
LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to
More informationIntermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
More informationUNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA
7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K Chpter Wht ou will lern 7Prols nd other grphs Eploring prols Skething prols with trnsformtions Skething prols using ftoristion Skething ompleting the squre Skething using
More informationTwo Triads of Congruent Circles from Reflections
Forum Geometriorum Volume 8 (2008) 7 12. FRUM GEM SSN 1534-1178 Two Trids of ongruent irles from Refletions Qung Tun ui strt. Given tringle, we onstrut two trids of ongruent irles through the verties,
More informationMCH T 111 Handout Triangle Review Page 1 of 3
Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:
More informationILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS
ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu
More informationGeometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.
Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution
More information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
More informationLearning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
More informationPYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS
PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More informationUNCORRECTED. Australian curriculum NUMBER AND ALGEBRA
0A 0B 0C 0D 0E 0F 0G 0H Chpter Wht ou will lern Qudrti equtions (Etending) Solving + = 0 nd = d (Etending) Solving + + = 0 (Etending) Applitions of qudrti equtions (Etending) The prol Skething = with diltions
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More informationy z A left-handed system can be rotated to look like the following. z
Chpter 2 Crtesin Coördintes The djetive Crtesin bove refers to René Desrtes (1596 1650), who ws the first to oördintise the plne s ordered pirs of rel numbers, whih provided the first sstemti link between
More information2.1 ANGLES AND THEIR MEASURE. y I
.1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the
More informationMathematics 10 Page 1 of 5 Properties of Triangle s and Quadrilaterals. Isosceles Triangle. - 2 sides and 2 corresponding.
Mthemtis 10 Pge 1 of 5 Properties of s Pthgoren Theorem 2 2 2 used to find the length of sides of right tringle Tpe of s nd Some s Theorems ngles s Slene Isoseles Equilterl ute - ll ngles re less thn 90
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationChapter17. Congruence and transformations. Contents: A Transformations B Congruent figures C Congruent triangles D Proof using congruence
hpter17 ongruene nd trnsfortions ontents: Trnsfortions ongruent figures ongruent tringles Proof using ongruene 352 ONGRUENE N TRNSFORMTIONS (hpter 17) Opening prole Jne ut two tringulr slies of heeseke,
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationNaming the sides of a right-angled triangle
6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship
More informationThis enables us to also express rational numbers other than natural numbers, for example:
Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers
More informationSymmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent
More informationLesson 4.1 Triangle Sum Conjecture
Lesson 4.1 ringle um onjeture me eriod te n erises 1 9, determine the ngle mesures. 1. p, q 2., 3., 31 82 p 98 q 28 53 17 79 23 50 4. r, s, 5., 6. t t s r 100 85 100 30 4 7 31 7. s 8. m 9. m s 76 35 m
More informationMATHEMATICS AND STATISTICS 1.6
MTHMTIS N STTISTIS 1.6 pply geometri resoning in solving prolems ternlly ssessed 4 redits S 91031 inding unknown ngles When finding the size of unknown ngles in figure, t lest two steps of resoning will
More information( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).
Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.
More informationGRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames
Visuliing Quternions Andrew J. Hnson Computer Siene Deprtment Indin Universit Siggrph Tutoril GRAND PLAN I: Fundmentls of Quternions II: Visuliing Quternion Geometr III: Quternion Frmes IV: Clifford Algers
More informationDrill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.
Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte
More informationTrigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
More informationFormula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx
Fill in the Blnks for the Big Topis in Chpter 5: The Definite Integrl Estimting n integrl using Riemnn sum:. The Left rule uses the left endpoint of eh suintervl.. The Right rule uses the right endpoint
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationVectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:
hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points
More informationProportions: A ratio is the quotient of two numbers. For example, 2 3
Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)
More informationCLASS XII. AdjA A. x X. a etc. B d
CLASS XII The syllus is divided into three setions A B nd C Setion A is ompulsory for ll ndidtes Cndidtes will hve hoie of ttempting questions from either Setion B or Setion C There will e one pper of
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationForces on curved surfaces Buoyant force Stability of floating and submerged bodies
Stti Surfe ores Stti Surfe ores 8m wter hinge? 4 m ores on plne res ores on urved surfes Buont fore Stbilit of floting nd submerged bodies ores on Plne res Two tpes of problems Horizontl surfes (pressure
More informationApplications of Definite Integral
Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
More informationare coplanar. ˆ ˆ ˆ and iˆ
SML QUSTION Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor ii The question pper onsists of 6 questions divided into three Setions, B nd C iii Question No
More informationStandard Trigonometric Functions
CRASH KINEMATICS For ngle A: opposite sine A = = hypotenuse djent osine A = = hypotenuse opposite tngent A = = djent For ngle B: opposite sine B = = hypotenuse djent osine B = = hypotenuse opposite tngent
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationUNIT 31 Angles and Symmetry: Data Sheets
UNIT 31 Angles nd Symmetry Dt Sheets Dt Sheets 31.1 Line nd Rottionl Symmetry 31.2 Angle Properties 31.3 Angles in Tringles 31.4 Angles nd Prllel Lines: Results 31.5 Angles nd Prllel Lines: Exmple 31.6
More informationSimilar Right Triangles
Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems
More informationMathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers
Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule
More informationSection 3.6. Definite Integrals
The Clulus of Funtions of Severl Vribles Setion.6 efinite Integrls We will first define the definite integrl for funtion f : R R nd lter indite how the definition my be extended to funtions of three or
More informationHS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem
HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More informationBEGINNING ALGEBRA (ALGEBRA I)
/0 BEGINNING ALGEBRA (ALGEBRA I) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: http://www.glendle.edu/studypkets Students who hve tken yer of high shool lger or its equivlent with grdes
More informationA quick overview of the four conic sections in rectangular coordinates is presented below.
MAT 6H Rectngulr Equtions of Conics A quick overview of the four conic sections in rectngulr coordintes is presented elow.. Circles Skipped covered in previous lger course.. Prols Definition A prol is
More informationTrigonometry. Trigonometry. labelling conventions. Evaluation of areas of non-right-angled triangles using the formulas A = 1 ab sin (C )
8 8 Pythgors theorem 8 Pythgoren trids 8 Three-dimensionl Pythgors theorem 8D Trigonometri rtios 8E The sine rule 8F miguous se of the sine rule 8G The osine rule 8H Speil tringles 8I re of tringles res
More informationk ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
More information4. Statements Reasons
Chpter 9 Answers Prentie-Hll In. Alterntive Ativity 9-. Chek students work.. Opposite sides re prllel. 3. Opposite sides re ongruent. 4. Opposite ngles re ongruent. 5. Digonls iset eh other. 6. Students
More informationSection - 2 MORE PROPERTIES
LOCUS Section - MORE PROPERTES n section -, we delt with some sic properties tht definite integrls stisf. This section continues with the development of some more properties tht re not so trivil, nd, when
More informationPROPERTIES OF TRIANGLES
PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties.
More informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More informationUsing technology to teach and solve challenging math problems
Using tehnology to teh nd solve hllenging mth prolems Ellin Grigoriev, PhD Astrt Dr. Ellin Grigoriev is Professor of Mthemtis t Tes Womn's University (TWU). he grduted from Mosow tte University first in
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls
More informationMATH Final Review
MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out
More informationThe Relationship between the Nine-Point Circle, the Circumscribed Circle and the Inscribed Circle of Archimedes Triangle Yuporn Rimcholakarn
Nresun Universit Journl: Siene nd Tehnolog 08; 63 The Reltionship between the Nine-Point irle, the irusribed irle nd the Insribed irle of rhiedes Tringle Yuporn Riholkrn Fult of Siene nd Tehnolog, Pibulsongkr
More information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
More informationExercise sheet 6: Solutions
Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd
More information