Part I: Study the theorem statement.

Size: px
Start display at page:

Download "Part I: Study the theorem statement."

Transcription

1 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 your group t the end of lss. Theorem (Pythgors Theorem). For right-ngled tringle the squre of the hypotenuse is equl to the sum of the squres of the other sides. Prt I: Study the theorem sttement. A. Restte the theorem. Restte the theorem s n If..., then... sttement, using vrile nmes to e speifi, nd expliitly identify the ssumptions nd onlusions. Wht re the ssumptions? Wht re the onlusions? Resttement of the theorem: B. Drw piture. Illustrte the ssumptions of the theorem with piture, leling using the sme vrile nmes tht you used ove. 1

2 2 A STUDY OF PYTHAGORAS THEOREM C. More with pitures nd exmples One lssi exmple of right tringle is Explin how the following piture illustrtes the theorem sttement:. Give nother exmple of right tringle with integer side lengths, nd drw the orresponding piture. Give n exmple of right tringle with t lest one non-integer-length side. Give n exmple of right tringle with no sides of integer length. D. Apply the theorem to trivil nd extreme exmples. Wht does the theorem sttement sy out right tringles where one of the legs of the tringle is ritrrily smll (very very lose to 0 ompred to the other leg)?

3 A STUDY OF PYTHAGORAS THEOREM 3 E. Rte the strength of the ssumptions nd onlusions. Give n exmple of slightly weker sttement. Give n exmple of relted question tht would require slightly stronger sttement. F. Compre with other theorems Look up nd stte the Lw of Cosines nd ompre the theorem sttements. G. Oserve the detil. Wht kinds of tringles does this theorem tell you out? Wht re some exmples of tringles tht this theorem doesn t pply to? H. Clssify wht the theorem does nd how it n e used. Conretely, wht does this theorem enle you to do?

4 4 A STUDY OF PYTHAGORAS THEOREM I. Chek the onverse Stte the onverse of Pythgors Theorem. Use the Lw of Cosines to verify tht the onverse is true. Exmple: Suppose tht we hve tringle with sides of length 2, 2.1, nd 2.9. We hve (2.1) 2 = (2.9) 2. The tringle is therefore right-ngled. Question: Did we use Pythgors Theorem or its onverse to dedue this? Explin. Exmple: Suppose tht we hve tringle with sides of length 3.6, 7.7, nd 8.4. We hve (6.6) 2 + (7.7) 2 = (8.4) 2. The tringle is therefore not right-ngled. Question: Did we use Pythgors Theorem or its onverse to dedue this? Explin.

5 A STUDY OF PYTHAGORAS THEOREM 5 Prt II: Proof of Pythgors Theorem. A. Exmples versus proofs. We hve hve done some exmples of Pythgors Theorem. Why doesn t this reple proving Pythgors Theorem? Explin why the piture in prt I.C (with the oxes) is lso not proof of Pythgors Theorem. Now let s onsider the following proof of Pythgors Theorem. Proof. The proof n e shown using the two squres in Figure 1. To drw the first squre egin y drwing generl right tringle with sides of length nd, nd emedding it in the lower-left orner of squre with side lengths +. Then rotte the originl tringle y 90 t time, pling one opy in eh of the other orners of the squre to get the left-most piture in Figure 1. Then we drw nother ( + ) ( + ) squre, nd one gin ple four opies of the originl tringle inside this squre; ut this time, we pir them up to mke nd retngles, nd ple them in the upper-left nd lower-right orners, respetively. This gives the right-most piture in Figure 1. Now, we ompute the re of the ( + ) ( + ) squre in two wys. Nmely, the re of the left squre is 2 + (4 Are of (, )-tringle), nd the re of the right squre is So 2 = 2 + 2, s desired (4 Are of (, )-tringle). Figure 1. Geometri proof of Pythgors Theorem

6 6 A STUDY OF PYTHAGORAS THEOREM Find where the ssumptions re used. Where did we use the ft tht the originl tringle ws right tringle? Chek the text. How would the proof hd roken if the originl tringle hdn t een right tringle? The proof ws missing step! The piture implies tht the white interior in the left-most piture in Figure 1 ws squre, ut this ws not tully verified! Use some geometry tht the interior ngles of tht white shpe re indeed right ngles now. Chek the proof here ginst the proof given in the ook. Cn you find other missing steps or ssumptions in the ook s proof?

1 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. 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 information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS 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 information

A study of Pythagoras Theorem

A 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 information

GM1 Consolidation Worksheet

GM1 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 information

THE PYTHAGOREAN THEOREM

THE 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 information

Maintaining Mathematical Proficiency

Maintaining 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 information

Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: 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 information

Comparing the Pre-image and Image of a Dilation

Comparing 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 information

Proving the Pythagorean Theorem

Proving 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 information

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths

Intermediate 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 information

Section 1.3 Triangles

Section 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

Trigonometry and Constructive Geometry

Trigonometry 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 information

MTH 4-16a Trigonometry

MTH 4-16a Trigonometry MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

More information

12.4 Similarity in Right Triangles

12.4 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 information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 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 information

Triangles The following examples explore aspects of triangles:

Triangles The following examples explore aspects of triangles: Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

More information

= x x 2 = 25 2

= x x 2 = 25 2 9.1 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 7, 2016 Geometry 9.1 The Pythgoren Theorem 1 Geometry 9.1 The Pythgoren Theorem 9.1

More information

Trigonometry Revision Sheet Q5 of Paper 2

Trigonometry 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 information

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

In 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 information

5. Every rational number have either terminating or repeating (recurring) decimal representation.

5. 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 information

9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1

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 information

Lesson 2.1 Inductive Reasoning

Lesson 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 information

Non Right Angled Triangles

Non 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 information

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.

3 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 information

Bases for Vector Spaces

Bases for Vector Spaces Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

More information

Pythagoras Theorem. Pythagoras

Pythagoras Theorem. Pythagoras 11 Pythgors Theorem Pythgors Theorem. onverse of Pythgors theorem. Pythgoren triplets Proof of Pythgors theorem nd its converse Problems nd riders bsed on Pythgors theorem. This unit fcilittes you in,

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

Activities. 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

Activities. 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 information

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving right-ngle tringles Sine nd osine rule Lelling right-ngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

Similarity and Congruence

Similarity 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 information

6.2 The Pythagorean Theorems

6.2 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 information

Lesson 2.1 Inductive Reasoning

Lesson 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 information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 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 information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

Is there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them!

Is 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 information

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides

Pythagoras 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 information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

An Overview of Integration

An Overview of Integration An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 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 information

Shape and measurement

Shape 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 information

LESSON 11: TRIANGLE FORMULAE

LESSON 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 information

Definition :- A shape has a line of symmetry if, when folded over the line. 1 line of symmetry 2 lines of symmetry

Definition :- A shape has a line of symmetry if, when folded over the line. 1 line of symmetry 2 lines of symmetry Symmetry Lines of Symmetry Definition :- A shpe hs line of symmetry if, when folded over the line the hlves of the shpe mtch up exctly. Some shpes hve more thn one line of symmetry : line of symmetry lines

More information

8Similarity 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.

8Similarity 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 information

Green 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. (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 information

Number systems: the Real Number System

Number systems: the Real Number System Numer systems: the Rel Numer System syllusref eferenceence Core topic: Rel nd complex numer systems In this ch chpter A Clssifiction of numers B Recurring decimls C Rel nd complex numers D Surds: suset

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 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 information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their

More information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

Section 2.1 Special Right Triangles

Section 2.1 Special Right Triangles Se..1 Speil Rigt Tringles 49 Te --90 Tringle Setion.1 Speil Rigt Tringles Te --90 tringle (or just 0-60-90) is so nme euse of its ngle mesures. Te lengts of te sies, toug, ve very speifi pttern to tem

More information

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

List 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 information

2 Calculate the size of each angle marked by a letter in these triangles.

2 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 information

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION 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

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

Identifying and Classifying 2-D Shapes

Identifying 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 information

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS

PYTHAGORAS 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 information

Math Lesson 4-5 The Law of Cosines

Math 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 information

HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem

HS 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 information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

THE 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 information

Introduction To Matrices MCV 4UI Assignment #1

Introduction To Matrices MCV 4UI Assignment #1 Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

Plotting Ordered Pairs Using Integers

Plotting 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 information

p-adic Egyptian Fractions

p-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 information

8Similarity ONLINE 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.

8Similarity ONLINE 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 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step

More information

PHYS 1114, Lecture 1, January 18 Contents:

PHYS 1114, Lecture 1, January 18 Contents: PHYS 1114, Lecture 1, Jnury 18 Contents: 1 Discussed Syllus (four pges). The syllus is the most importnt document. You should purchse the ExpertTA Access Code nd the L Mnul soon! 2 Reviewed Alger nd Strted

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

CHENG Chun Chor Litwin The Hong Kong Institute of Education

CHENG 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 information

SYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus

SYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is

More information

Some Theory of Computation Exercises Week 1

Some Theory of Computation Exercises Week 1 Some Theory of Computtion Exercises Week 1 Section 1 Deterministic Finite Automt Question 1.3 d d d d u q 1 q 2 q 3 q 4 q 5 d u u u u Question 1.4 Prt c - {w w hs even s nd one or two s} First we sk whether

More information

1.3 SCALARS AND VECTORS

1.3 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 information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 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 information

Similar Right Triangles

Similar 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 information

Objective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15

Objective: 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 information

More on automata. Michael George. March 24 April 7, 2014

More on automata. Michael George. March 24 April 7, 2014 More on utomt Michel George Mrch 24 April 7, 2014 1 Automt constructions Now tht we hve forml model of mchine, it is useful to mke some generl constructions. 1.1 DFA Union / Product construction Suppose

More information

Pythagoras theorem and surds

Pythagoras 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 information

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the

More information

Polynomials. Polynomials. Curriculum Ready ACMNA:

Polynomials. Polynomials. Curriculum Ready ACMNA: Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw

More information

Figure 1. The left-handed and right-handed trefoils

Figure 1. The left-handed and right-handed trefoils The Knot Group A knot is n emedding of the irle into R 3 (or S 3 ), k : S 1 R 3. We shll ssume our knots re tme, mening the emedding n e extended to solid torus, K : S 1 D 2 R 3. The imge is lled tuulr

More information

How 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?

How 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 information

Lecture 3: Equivalence Relations

Lecture 3: Equivalence Relations Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts

More information

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6 CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

Find the value of x. Give answers as simplified radicals.

Find 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 information

Mathematics Number: Logarithms

Mathematics Number: Logarithms plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 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 information

Consolidation Worksheet

Consolidation Worksheet Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

More information

JURONG JUNIOR COLLEGE

JURONG JUNIOR COLLEGE JURONG JUNIOR COLLEGE 2010 JC1 H1 8866 hysics utoril : Dynmics Lerning Outcomes Sub-topic utoril Questions Newton's lws of motion 1 1 st Lw, b, e f 2 nd Lw, including drwing FBDs nd solving problems by

More information

Symmetrical Components 1

Symmetrical 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 information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus 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 information

The Legacy of Pythagoras Theorem

The 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 information

Linear Inequalities. Work Sheet 1

Linear 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 information

Parallel Projection Theorem (Midpoint Connector Theorem):

Parallel Projection Theorem (Midpoint Connector Theorem): rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length one-hlf the third side. onversely, If line isects

More information

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( ) UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

More information