= x x 2 = 25 2
|
|
- Adam Scott
- 5 years ago
- Views:
Transcription
1 9.1 Wrm Up Solve the eqution = x x 2 = x 2 = x 2 = 12 2 Mrh 7, 2016 Geometry 9.1 The Pythgoren Theorem 1
2 Geometry 9.1 The Pythgoren Theorem
3 9.1 Essentil Question How n I use the Pythgoren Theorem to find lengths of the sides of Right Tringles? Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 3
4 Gols Prove the Pythgoren Theorem. Solve tringles using the theorem. Solve prolems using the theorem. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 4
5 This is nient history. The Egyptin Pyrmid uilders used it to mke squre orners. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 5
6 Leg Terminology The two legs form the right ngle. The hypotenuse is ross from the right ngle. Leg Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 6
7 Hypotenuse = strethed ginst Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 7
8 Proof Proofs of the Pythgoren Theorem re numerous well over 300 known. Disovered in mny nient ultures. Eulid s is mong the most diffiult to understnd. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 8
9 Eulid s Digrm Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 9
10 Chinese Proof Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 10
11 Chinese Proof Prt 1: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 11 Are of the smll squre: A = 2 Are of one tringle: A = (½) Are of 4 tringles: A = 2
12 Chinese Proof Prt 1: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 12 Are of the smll squre: A = 2 Are of 4 tringles: A = 2 Are Sum 2 + 2
13 ? Chinese Proof Prt 2: + Are Sum Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 13? +
14 Chinese Proof Prt 2: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 14 Are Sum Are of the lrge squre: A ( ) ( )( )
15 Chinese Proof Prt 3: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 15 Are Sum or These res re equl.
16 Chinese Proof Prt 3: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 16
17 Proof y President Grfield (1876) 20 th President of the United Sttes Are of Trpezoid = Sum of re of three tringles Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 17
18 Pythgoren Theorem In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. If ABC is right tringle, then = 2 B Mrh 7, 2016 C Geometry 9.3 Converse of the Pythgoren Theorem A 18
19 Exmple 1 Solve Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 19
20 Exmple 2 Solve Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 20
21 x Exmple 3 20 x Solve x x 400 x 200 x x 200 x 10 2 x Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 21
22 Your Turn Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 22
23 Pythgoren Triples nd re Pythgoren Triples. Eh side is n integer. A Pythgoren Triple multiplied y whole numer results in nother Pythgoren Triple Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 23
24 Pythgoren Triples Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 24
25 Exmple 4 Is Pythgoren Triple? = 20 2? Flse! Not Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 25
26 Exmple 5 Is Pythgoren Triple? = 29 2? = 841? 841 = 841 True It is Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 26
27 Exmple 6 The distne etween ses on sell dimond is 90 feet. A ther throws the ll from home se to 2 nd se. Wht is the distne? Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 27
28 Exmple Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 28
29 Exmple in in. Find the digonl mesure of the LCD sreen to the nerest inh. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 29
30 Exmple in in. Find the digonl mesure of the LCD sreen to the nerest inh. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 30
31 Exmple 7 Find the digonl mesure of the LCD sreen to the nerest inh in in Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 31
32 Exmple 7 Find the digonl mesure of the LCD sreen to the nerest inh in in Aout 42.2 inhes Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 32
33 Dy 1 Summry In right tringle, the hypotenuse is the longest side. The sum of the squres of the legs is equl to the squre of the hypotenuse. If the three sides re ll integers, they form Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 33
34 True or Flse? The sum of the squre roots of ny two sides of n isoseles tringle is equl to the squre root of the remining side. Oh joy! Rpture! I got rin! + =? Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 34
35 Flse. It should hve een The sum of the squres of the two legs of right tringle is equl to the squre of the remining side. Oh joy! Rpture! I hve rin! Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 35
36 Gols Determine if tringle is right tringle. Use the Pythgoren inequlities to determine if tringle is ute or otuse. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 36
37 Pythgoren Theorem In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. If ABC is right tringle, then = 2 B C Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 37 A
38 Converse of Pythgoren Theorem If the squre of the length of the longest side of tringle is equl to the sum of the squres of the lengths of the other two sides, then the tringle is right tringle. If = 2, then ABC is right tringle. B C Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 38 A
39 Exmple 8 Is POD right tringle? 2 2? O ? P 34 Longest Side D Yes! Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 39
40 Exmple 9 Is SAD right tringle? 9 S 20 Whih segment is the longest? SD 2 2? ? A 12 No! D Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 40
41 Reminder x 2 = x 3x 2 = 3 2 x 2 = 9x = 5 3 x 2 = 3 2 x 2 = 9x 17 2 = = = 9 3 = 27 3x 2 = 3x = = 16 5 = 80 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 41
42 Exmple 10 Is HUG right tringle? H Whih segment is the longest? HG ? ? U 10 Yes! G? Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 42
43 Your Turn. 10 Is RST right? S ? ? R 26 T Yes it is. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 43
44 Tringle Inequlity Theorem In tringle, the sum of ny two sides is greter thn the third side > > > 4 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 44
45 Tringle Inequlity Theorem 5 4 This is not tringle sine < Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 45
46 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 46
47 Begin with right tringle = 2 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 47
48 Rotte side in. 2 = < nd hve not hnged hs not hnged. got smller. 2 got smller. nd The right ngle gets smller: it is ute. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 48
49 Theorem 9.6 If the squre of the length of the longest side of tringle is less thn the sum of the squres of the other two sides, then the tringle is ute. A 2 < C B Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 49
50 Tke nother right tringle = 2 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 50
51 Rotte side out. 2 = > nd hve not hnged hs not hnged. got lrger. 2 got lrger. nd The right ngle gets lrger: it is otuse. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 51
52 Theorem 9.6 If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the other two sides, then the tringle is otuse. A 2 > C B Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 52
53 Exmple 11 The sides of tringle mesure 5, 7, nd 11. Clssify it s ute, right, or otuse. Solution: The longest side is ? ? > 74 Otuse Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 53
54 Exmple 12 The sides of tringle re 17, 20, nd 25. Clssify the tringle. Solution: 25 2? ? < 689 Aute Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 54
55 Exmple 13 Clssify this tringle ? ? Right Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 55
56 Exmple 14 Clssify this tringle It isn t tringle! 6 +8 < 16. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 56
57 Summry If 2 = 2 + 2, RIGHT. If 2 < 2 + 2, ACUTE. If 2 > 2 + 2, OBTUSE. The lst two n e very onfusing; don t get them mixed up. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 57
58 Homework Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 58
59 Generting Pythgoren Triples Find two positive integers & whih re reltively prime nd >. Tht is, they hve no ftors in ommon other thn 1. Then the triples re: 2 + 2, 2 nd 2 2. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 59
60 Generting Pythgoren Triples Exmple: Choose = 4 nd = = = = 2(4)(3) = = = 7. 7, 24, 25 is Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 60
61 Generting Pythgoren Triples 7, 24, 25 is Pythgoren Triple. Chek: = 25 2? = 625? 625 = 625 Tht s triple! Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 61
62 Pythgoren Triples nd re reltively prime. > Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 62
63 Try it. Using = 8 nd = 3, find the Pythgoren Triple. Answer: = 64 9 = 55 2(8)(3) = = = 73 2? = 5329? 5329 = 5329 heks. Are Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 63
9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1
9.1 Dy 1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 1, 2017 Geometry 9.1 The Pythgoren Theorem 1 9.1 Dy 2 Wrm Up Use the Pythgoren
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 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 informationProving the Pythagorean Theorem. Proving the Pythagorean Theorem and the Converse of the Pythagorean Theorem
.5 Proving the Pythgoren Theorem Proving the Pythgoren Theorem nd the Converse of the Pythgoren Theorem Lerning Gols In this lesson, you will: Prove the Pythgoren Theorem using similr tringles. Prove the
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 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 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 informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
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 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 information6.2 The Pythagorean Theorems
PythgorenTheorems20052006.nb 1 6.2 The Pythgoren Theorems One of the best known theorems in geometry (nd ll of mthemtics for tht mtter) is the Pythgoren Theorem. You hve probbly lredy worked with this
More informationLesson 2.1 Inductive Reasoning
Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,
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 informationPart I: Study the theorem statement.
Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for
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 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 informationTrigonometry and Constructive Geometry
Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties
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 informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
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 informationComputing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt
Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2:
More informationMath 154B Elementary Algebra-2 nd Half Spring 2015
Mth 154B Elementry Alger- nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationLesson 2.1 Inductive Reasoning
Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,
More informationIdentifying and Classifying 2-D Shapes
Ientifying n Clssifying -D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte
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 informationIn right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
Mth 3329-Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then
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 informationMATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 1 MTH SPEK - TO E UNDERSTOOD ND MEMIZED 1) TRINGLE = 2-dimentionl she hving 3 sides nd 3 ngles. HRTERISTI OF TRINGLES I) Every tringle is n enclosed she tht hs these
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 informationTHE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES
THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,
More informationMath Lesson 4-5 The Law of Cosines
Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude
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 informationFind the value of x. Give answers as simplified radicals.
9.2 Dy 1 Wrm Up Find the vlue of. Give nswers s simplified rdicls. 1. 2. 3 3 3. 4. 10 Mrch 2, 2017 Geometry 9.2 Specil Right Tringles 1 Geometry 9.2 Specil Right Tringles 9.2 Essentil Question Wht is the
More informationIs there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them!
PUSHING PYTHAGORAS 009 Jmes Tnton A triple of integers ( bc,, ) is clled Pythgoren triple if exmple, some clssic triples re ( 3,4,5 ), ( 5,1,13 ), ( ) fond of ( 0,1,9 ) nd ( 119,10,169 ). + b = c. For
More informationPythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.
Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45 www.mthletis.om Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived
More information1 cos. cos cos cos cos MAT 126H Solutions Take-Home Exam 4. Problem 1
MAT 16H Solutions Tke-Home Exm 4 Problem 1 ) & b) Using the hlf-ngle formul for cosine, we get: 1 cos 1 4 4 cos cos 8 4 nd 1 8 cos cos 16 4 c) Using the hlf-ngle formul for tngent, we get: cot ( 3π 1 )
More informationA study of Pythagoras Theorem
CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins
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 informationPythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides
Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the
More informationPlotting Ordered Pairs Using Integers
SAMPLE Plotting Ordered Pirs Using Integers Ple two elsti nds on geoord to form oordinte xes shown on the right to help you solve these prolems.. Wht letter of the lphet does eh set of pirs nme?. (, )
More informationSimilarity and Congruence
Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
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 informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
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 informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
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 informationObjective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15
Geometry CP Lesson 8.2 Pythgoren Theorem nd its Converse Pge 1 of 2 Ojective: Use the Pythgoren Theorem nd its converse to solve right tringle prolems. CA Geometry Stndrd: 12, 14, 15 Historicl Bckground
More informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationSpecial Numbers, Factors and Multiples
Specil s, nd Student Book - Series H- + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book - Series H Contents Topics Topic - Odd, even, prime nd composite numers Topic - Divisiility tests
More informationLog1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?
008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing
More information(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
More information2 Calculate the size of each angle marked by a letter in these triangles.
Cmridge Essentils Mthemtics Support 8 GM1.1 GM1.1 1 Clculte the size of ech ngle mrked y letter. c 2 Clculte the size of ech ngle mrked y letter in these tringles. c d 3 Clculte the size of ech ngle mrked
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 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 information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
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 informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More information3.1 Review of Sine, Cosine and Tangent for Right Angles
Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,
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 informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationS2 (2.2) Pythagoras.notebook March 04, 2016
S2 (2.2) Pythgors.noteook Mrh 04, 2016 Dily Prtie 16.12.2015 Q1. Multiply out nd simplify 9x 3(2x + 1) Q2. Solve the eqution 3(2x + 4) = 18 Q3. If 1 = $1.30, how muh is 50 in dollrs? Tody we will e lerning
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 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 information03. Early Greeks & Aristotle
03. Erly Greeks & Aristotle I. Erly Greeks Topis I. Erly Greeks II. The Method of Exhustion III. Aristotle. Anximnder (. 60 B.C.) to peiron - the unlimited, unounded - fundmentl sustne of relity - underlying
More informationSomething found at a salad bar
Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOH-H-TO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationThe Legacy of Pythagoras Theorem
Prol Volue 39, Issue 1(2003) The Legy of Pythgors Theore Peter G.rown 1 When sked wht thetil result they reeer fro High Shool, the verge person would proly reply with Pythgors Theore. In ny right tringle,
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 informationTorsion in Groups of Integral Triangles
Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,
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 informationLecture 2 : Propositions DRAFT
CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte
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 information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationMath 7, Unit 9: Measurement: Two-Dimensional Figures Notes
Mth 7, Unit 9: Mesurement: Two-Dimensionl Figures Notes Precision nd Accurcy Syllus Ojective: (6.) The student will red the pproprite mesurement tool to the required degree of ccurcy. We often use numers
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationSect 10.2 Trigonometric Ratios
86 Sect 0. Trigonometric Rtios Objective : Understnding djcent, Hypotenuse, nd Opposite sides of n cute ngle in right tringle. In right tringle, the otenuse is lwys the longest side; it is the side opposite
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More information42nd International Mathematical Olympiad
nd Interntionl Mthemticl Olympid Wshington, DC, United Sttes of Americ July 8 9, 001 Problems Ech problem is worth seven points. Problem 1 Let ABC be n cute-ngled tringle with circumcentre O. Let P on
More informationGEOMETRY OF THE CIRCLE TANGENTS & SECANTS
Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS www.mthletics.com.u Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the
More informationPythagoras theorem and surds
HPTER Mesurement nd Geometry Pythgors theorem nd surds In IE-EM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of right-ngled tringle. This result is known s Pythgors
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 informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationIndividual Contest. English Version. Time limit: 90 minutes. Instructions:
Elementry Mthemtics Interntionl Contest Instructions: Individul Contest Time limit: 90 minutes Do not turn to the first pge until you re told to do so. Write down your nme, your contestnt numer nd your
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More information50 AMC Lectures Problem Book 2 (36) Substitution Method
0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem
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 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 information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
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 informationAlg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A
lg 3 h 7.2, 8 1 7.2 Right Tringle Trig ) Use of clcultor sin 10 = sin x =.4741 c ) rete right tringles π 1) If = nd = 25, find 6 c 2) If = 30, nd = 45, = 1 find nd c 3) If in right, with right ngle t,
More informationPythagorean Theorem and Trigonometry
Ptgoren Teorem nd Trigonometr Te Ptgoren Teorem is nient, well-known, nd importnt. It s lrge numer of different proofs, inluding one disovered merin President Jmes. Grfield. Te we site ttp://www.ut-te-knot.org/ptgors/inde.stml
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 informationAnswers to Exercises. c 2 2ab b 2 2ab a 2 c 2 a 2 b 2
Answers to Eercises CHAPTER 9 CHAPTER LESSON 9. CHAPTER 9 CHAPTER. c 9. cm. cm. b 5. cm. d 0 cm 5. s cm. c 8.5 cm 7. b cm 8.. cm 9. 0 cm 0. s.5 cm. r cm. 7 ft. 5 m.. cm 5.,, 5. 8 m 7. The re of the lrge
More information8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.
8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims
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 information