Proving Lines Parallel

Size: px
Start display at page:

Download "Proving Lines Parallel"

Transcription

1 rovig Lies arallel ecogize agle coditios that occur with parallel lies. rove that two lies are parallel based o give agle relatioships. do you kow that the sides of a parkig space are parallel? Have you ever bee i a tall buildig ad looked dow at a parkig lot? The parkig lot is full of lie segets that appear to be parallel. The workers who pait these lies ust be certai that they are parallel. IDETIFY LLEL LIE Whe each stripe of a parkig space itersects the ceter lie, the agles fored are correspodig agles. If the lies are parallel, we kow that the correspodig agles are cogruet. Coversely, if the correspodig agles are cogruet, the the lies ust be parallel. ostulate. If two lies i a plae are cut by a trasversal so that correspodig agles are cogruet, the the lies are parallel. bbreviatio: If corr. are, the lies are. Eaples: If 5, 6,, or 8, the ostulate. justifies the costructio of parallel lies. arallel Lie Through a oit ot o Lie tudy Tip Look Back To review copyig agles, see Lesso -. Use a straightedge to draw a lie. Label two poits o the lie as ad. Draw a poit that is ot o. Draw. Copy so that is the verte of the ew agle. Label the itersectio poits ad. Draw. Because by costructio ad they are correspodig agles,. Lesso -5 rovig Lies arallel 5 David ailors/cobi

2 The costructio establishes that there is at least oe lie through that is parallel to. I 95, cottish physicist ad atheaticia Joh layfair provided the oder versio of Euclid s arallel ostulate, which states there is eactly oe lie parallel to a lie through a give poit ot o the lie. ostulate.5 arallel ostulate If give a lie ad a poit ot o the lie, the there eists eactly oe lie through the poit that is parallel to the give lie. arallel lies with a trasversal create ay pairs of cogruet agles. Coversely, those pairs of cogruet agles ca deterie whether a pair of lies is parallel. Theores Eaples.5 If two lies i a plae are cut by a trasversal so that If 8 or if a pair of alterate eterior agles is cogruet, the, the. the two lies are parallel. bbreviatio: If alt. et. are, the lies are..6 If two lies i a plae are cut by a trasversal so that If 5 80 a pair of cosecutive iterior agles is suppleetary, or if 6 the the lies are parallel. 80, the. bbreviatio: If cos. it. are suppl., the lies are.. If two lies i a plae are cut by a trasversal so that a If 6 or if pair of alterate iterior agles is cogruet, the the 5, the. lies are parallel. bbreviatio: If alt. it. are, the lies are..8 I a plae, if two lies are perpedicular to the sae lie, If ad, the they are parallel. the. bbreviatio: If lies are to the sae lie, the lies are. rovig Lies arallel Eaple Idetify arallel Lies I the figure, BG bisects BH. Deterie which lies, if ay, are parallel. The su of the agle easures i a triagle ust be 80, so BDF 80 (5 65) or 0. ice BDF ad BGH have the sae easure, they are cogruet. Cogruet correspodig agles idicate parallel lies. o, DF GH. BD DBF, because BG bisects BH. o, BD 5. BD ad BDF are alterate iterior agles, but they have differet easures so they are ot cogruet. Thus, B is ot parallel to DF or GH. G D 0 B 5 65 F H 5 Chapter arallel ad erpedicular Lies

3 gle relatioships ca be used to solve probles ivolvig ukow values. Eaple olve robles with arallel Lies LGEB Fid ad U so that. Eplore Fro the figure, you kow that U 8 ad TV 9. You also kow that U ad TV are correspodig agles. (8 ) U (9 ) T V la For lie to be parallel to lie, the correspodig agles ust be cogruet. o, U TV. ubstitute the give agle easures ito this equatio ad solve for. Oce you kow the value of, use substitutio to fid U. olve U TV Correspodig agles 8 9 ubstitutio ubtract 8 fro each side. 5 dd to each side. ow use the value of to fid U. U 8 Origial equatio 8(5) 5 iplify. Eaie Verify the agle easure by usig the value of to fid TV. That is, 9 9(5) or. ice U TV, U TV ad. tudy Tip rovig Lies arallel Whe provig lies parallel, be sure to check for cogruet correspodig agles, alterate iterior agles, alterate eterior agles, or suppleetary cosecutive iterior agles. OVE LIE LLEL The agle pair relatioships fored by a trasversal ca be used to prove that two lies are parallel. Eaple Give: rove: roof: r s 5 6 rove Lies arallel 5 6 tateets easos. r s, 5 6. Give. ad 5 are suppleetary.. Cosecutive Iterior gle Theore Defiitio of suppleetary agles Defiitio of cogruet agles ubstitutio roperty () 6. ad 6 are suppleetary. 6. Defiitio of suppleetary agles.. If cos. it. are suppl., the lies are. r s Lesso -5 rovig Lies arallel 5

4 I Lesso -, you leared that parallel lies have the sae slope. You ca use the slopes of lies to prove that lies are parallel. Eaple Deterie whether g f. lope ad arallel Lies slope of f: 0 or 6 0 slope of g: or 0 () ice the slopes are the sae, g f. g y (0, ) (6, ) (, 0) (, 0) O f Cocept Check. uarize five differet ethods to prove that two lies are parallel.. Fid a coutereaple for the followig stateet. If lies ad are cut by trasversal t so that cosecutive iterior agles are cogruet, the lies ad are parallel ad t is perpedicular to both lies.. OE EDED Describe two situatios i your ow life i which you ecouter parallel lies. How could you verify that the lies are parallel? Guided ractice Give the followig iforatio, deterie which lies, if ay, are parallel. tate the postulate or theore that justifies your aswer Fid so that (5 90) (9 5) 6 5 ( ) p q ( 9) 0. OOF Write a two-colu proof of Theore.5.. Deterie whether p q. C(, ) 6 8 O 6 B(0, ) (, ) 6 y D(6, ) p q pplicatio. HYIC The Hubble Telescope gathers parallel light rays ad directs the to a cetral focal poit. Use a protractor to easure several of the agles show i the diagra. re the lies parallel? Eplai how you kow. 5 Chapter arallel ad erpedicular Lies

5 ractice ad pply For Eercises 6 5, 8 9 ee Eaples Etra ractice ee page 60. Give the followig iforatio, deterie which lies, if ay, are parallel. tate the postulate or theore that justifies your aswer.. 8 a b EF BFG 8. EB DBC 9. EFB CBF 0. GFD CBD E 6 5 B C D F G. HLK JL. L L. L L 80. H, JT K H L J T Latitude lies are parallel, ad logitude lies appear parallel i certai locatios o Earth. Visit co/webquest to cotiue work o your Webuest project. 5. OOF Copy ad coplete the proof of Theore.8. Give: t t rove: roof: tateets easos. t, t.?. ad are right agles..?..?..? t Fid so that (9 ) 0 (8 ) (9 ) ( ) ( 5) ( 00) (5 90) ( 9). OOF Write a two-colu proof of Theore.6.. OOF Write a paragraph proof of Theore.. (8 ) ( 8) Lesso -5 rovig Lies arallel 55

6 OOF Write a two-colu proof for each of the followig.. Give: 5. Give: D CD rove: T UV rove: BC CD V C D W B T U 6. Give: J K. Give: ad are suppleetary. rove: K L rove: J K L Deterie whether each pair of lies is parallel. Eplai why or why ot. 8. y 9. B(, ) (, ) D(, ) y D(0,.5) C(.5,.8) O C(0, 0) (, 0.5) O B(,.5) Joh layfair I 95, Joh layfair published his versio of Euclid s Eleets. I his editio, layfair stadardized the otatio used for poits ad figures ad itroduced algebraic otatio for use i proofs. ource: athworld.wolfra.co 0. HOE IOVEET To build a fece, Ji positioed the fece posts ad the placed a board at a agle betwee the fece posts. s he placed each picket, he easured the agle that the picket ade with the. Why does this esure that the pickets will be parallel?. FOOTBLL Whe stripig the practice football field, r. Hawkiso first paited the sidelies. et he arked off 0-yard icreets o oe sidelie. He the costructed lies perpedicular to the sidelies at each 0-yard ark. Why does this guaratee that the 0-yard lies will be parallel? pickets. CITICL THIKIG Whe deel was workig o a art project, he drew a four-sided figure with two pairs of opposite parallel sides. He oticed soe patters relatig to the agles i the figure. List as ay patters as you ca about a -sided figure with two pairs of opposite parallel sides. fece posts board 56 Chapter arallel ad erpedicular Lies Brow Brothers. EECH Use the Iteret or other resource to fid atheaticias like Joh layfair who discovered ew cocepts ad proved ew theores related to parallel lies. Briefly describe their discoveries.

7 tadardized Test ractice. WITIG I TH swer the questio that was posed at the begiig of the lesso. How do you kow that the sides of a parkig space are parallel? Iclude the followig i your aswer: a copariso of the agles at which the lies forig the edges of a parkig space strike the ceterlie, ad a descriptio of the type of parkig spaces that for cogruet cosecutive iterior agles. 5. I the figure, lie is parallel to lie. Lie itersects both ad. Which of the followig lists icludes all of the agles that are suppleetary to? C agles,, ad B agles,, 6, ad agles, 5, ad 8 D agles,,, ad LGEB Kedra has at least oe quarter, oe die, oe ickel, ad oe pey. If she has three ties as ay peies as ickels, the sae uber of ickels as dies, ad twice as ay dies as quarters, the what is the least aout of oey she could have? $0. B $0.8 C $0.58 D $0.6 aitai Your kills ied eview Write a equatio i slope-itercept for for the lie that satisfies the give coditios. (Lesso -). 0., y-itercept is 6 8., cotais (, 5) 9. cotais (5, ) ad (, ) 50. perpedicular to y, cotais (, ) Fid the slope of each lie. (Lesso -) 5. BD 5. CD 5. B 5. EO 55. ay lie parallel to DE 56. ay lie perpedicular to BD y E(, ) B(0, ) O (, ) C(, ) D(, ) Gettig eady for the et Lesso Costruct a truth table for each copoud stateet. (Lesso -) 5. p ad q 58. p or q 59. p q 60. p q 6. CETY carpeter ust cut two pieces of wood at agles so that they fit together to for the corer of a picture frae. What type of agles ust he use to ake sure that a corer results? (Lesso -5) EEUIITE KILL Use the Distace Forula to fid the distace betwee each pair of poits. (To review the Distace Forula, see Lesso -.) 6. (, ), (, 9) 6. (8, 0), (, ) 6. (6, ), (8, ) Lesso -5 rovig Lies arallel 5

angles. Some pairs of the angles have special names. The following chart lists the pairs of Identify each pair of angles as alternate

angles. Some pairs of the angles have special names. The following chart lists the pairs of Identify each pair of angles as alternate Aswers (Lesso -) Lesso - - tud Guide ad Itervetio arallel Lies ad rasversals Relatioships Betwee Lies ad laes Whe two lies lie i the sae plae ad do ot itersect, the are parallel. Lies that do ot itersect

More information

Geometry Unit 3 Notes Parallel and Perpendicular Lines

Geometry Unit 3 Notes Parallel and Perpendicular Lines Review Cocepts: Equatios of Lies Geoetry Uit Notes Parallel ad Perpedicular Lies Syllabus Objective:. - The studet will differetiate aog parallel, perpedicular, ad skew lies. Lies that DO NOT itersect:

More information

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve

More information

Parallel and Perpendicular Lines

Parallel and Perpendicular Lines Parallel ad Perpedicular Lies. Pairs of Lies ad gles. Parallel Lies ad Trasversals. Proofs with Parallel Lies. Proofs with Perpedicular Lies.5 Equatios of Parallel ad Perpedicular Lies ike Path (p. ) rosswalk

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie

More information

9.5 Young s Double-Slit Experiment

9.5 Young s Double-Slit Experiment 9.5 Youg s Double-Slit Experiet Physics Tool box Early attepts to deostrate the iterferece of light were usuccessful because the two sources were too far apart ad out of phase, ad the wavelegth of light

More information

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1)

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1) Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU Thursda Ma, Review of Equatios of a Straight Lie (-D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio

More information

STRAIGHT LINES & PLANES

STRAIGHT LINES & PLANES STRAIGHT LINES & PLANES PARAMETRIC EQUATIONS OF LINES The lie "L" is parallel to the directio vector "v". A fixed poit: "( a, b, c) " o the lie is give. Positio vectors are draw from the origi to the fixed

More information

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009 Discrete Matheatics: Lectures 8 ad 9 Priciple of Iclusio ad Exclusio Istructor: Arijit Bishu Date: August ad 3, 009 As you ca observe by ow, we ca cout i various ways. Oe such ethod is the age-old priciple

More information

Null Spaces, Column. Transformations. Remarks. Definition ( ) ( ) Null A is the set of all x's in R that go to 0 in R 3

Null Spaces, Column. Transformations. Remarks. Definition ( ) ( ) Null A is the set of all x's in R that go to 0 in R 3 Null Spaces Colu Spaces ad Liear rasforatios earks Whe we study fuctios f ( ) we fid that a uderstadig of the doai D f itervals of - values ad the rage f itervals of y - values are ecessary to descrie

More information

3. Supppose the amount of information available on the web is multiplied by 27 every year. How much information will be available a.

3. Supppose the amount of information available on the web is multiplied by 27 every year. How much information will be available a. Lesso -A The Root of the Prole H ow would epoets work if the were fractios? To fid out, we we will use the iteret cocept, where the ase uer represets what the aout of iforatio gets ultiplied ever ear.

More information

Problem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =

Problem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t = Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,

More information

Integrals of Functions of Several Variables

Integrals of Functions of Several Variables Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio

More information

4.1 SIGMA NOTATION AND RIEMANN SUMS

4.1 SIGMA NOTATION AND RIEMANN SUMS .1 Sigma Notatio ad Riema Sums Cotemporary Calculus 1.1 SIGMA NOTATION AND RIEMANN SUMS Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each

More information

AVERAGE MARKS SCALING

AVERAGE MARKS SCALING TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I

More information

Lyman Memorial High School. Honors Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School. Honors Pre-Calculus Prerequisite Packet. Name: Lyma Memorial High School Hoors Pre-Calculus Prerequisite Packet 2018 Name: Dear Hoors Pre-Calculus Studet, Withi this packet you will fid mathematical cocepts ad skills covered i Algebra I, II ad Geometry.

More information

USA Mathematical Talent Search Round 3 Solutions Year 27 Academic Year

USA Mathematical Talent Search Round 3 Solutions Year 27 Academic Year /3/27. Fill i each space of the grid with either a or a so that all sixtee strigs of four cosecutive umbers across ad dow are distict. You do ot eed to prove that your aswer is the oly oe possible; you

More information

Calculus 2 Test File Fall 2013

Calculus 2 Test File Fall 2013 Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to

More information

Tennessee Department of Education

Tennessee Department of Education Teessee Departmet of Educatio Task: Comparig Shapes Geometry O a piece of graph paper with a coordiate plae, draw three o-colliear poits ad label them A, B, C. (Do ot use the origi as oe of your poits.)

More information

1. By using truth tables prove that, for all statements P and Q, the statement

1. By using truth tables prove that, for all statements P and Q, the statement Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3

More information

Department of Mathematics

Department of Mathematics WEEK Week 1 8 th Sept Departmet of Mathematics Scheme of Work Form 2 Term 12017-2018 Lesso Specific Objectives/Teachig Poits Number 1 Number Sets Natural umbers (N), Whole umbers (W), Itegers ( Z, Z, Z

More information

Define and Use Sequences and Series

Define and Use Sequences and Series . a., A..A; P..A, P..B TEKS Defie ad Use Sequeces ad Series Before You idetified ad wrote fuctios. Now You will recogize ad write rules for umber patters. Why? So you ca fid agle measures, as i Ex.. Key

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces

More information

Mathematics Extension 2

Mathematics Extension 2 009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard

More information

We will conclude the chapter with the study a few methods and techniques which are useful

We will conclude the chapter with the study a few methods and techniques which are useful Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs

More information

MODEL TEST PAPER II Time : hours Maximum Marks : 00 Geeral Istructios : (i) (iii) (iv) All questios are compulsory. The questio paper cosists of 9 questios divided ito three Sectios A, B ad C. Sectio A

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

More information

PROPERTIES OF AN EULER SQUARE

PROPERTIES OF AN EULER SQUARE PROPERTIES OF N EULER SQURE bout 0 the mathematicia Leoard Euler discussed the properties a x array of letters or itegers ow kow as a Euler or Graeco-Lati Square Such squares have the property that every

More information

Exponents. Learning Objectives. Pre-Activity

Exponents. Learning Objectives. Pre-Activity Sectio. Pre-Activity Preparatio Epoets A Chai Letter Chai letters are geerated every day. If you sed a chai letter to three frieds ad they each sed it o to three frieds, who each sed it o to three frieds,

More information

Academic. Grade 9 Assessment of Mathematics. Released assessment Questions

Academic. Grade 9 Assessment of Mathematics. Released assessment Questions Academic Grade 9 Assessmet of Mathematics 2014 Released assessmet Questios Record your aswers to the multiple-choice questios o the Studet Aswer Sheet (2014, Academic). Please ote: The format of this booklet

More information

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016 18th Bay Area Mathematical Olympiad February 3, 016 Problems ad Solutios BAMO-8 ad BAMO-1 are each 5-questio essay-proof exams, for middle- ad high-school studets, respectively. The problems i each exam

More information

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

More information

MATH spring 2008 lecture 3 Answers to selected problems. 0 sin14 xdx = x dx. ; (iv) x +

MATH spring 2008 lecture 3 Answers to selected problems. 0 sin14 xdx = x dx. ; (iv) x + MATH - sprig 008 lecture Aswers to selected problems INTEGRALS. f =? For atiderivatives i geeral see the itegrals website at http://itegrals.wolfram.com. (5-vi (0 i ( ( i ( π ; (v π a. This is example

More information

Solving equations (incl. radical equations) involving these skills, but ultimately solvable by factoring/quadratic formula (no complex roots)

Solving equations (incl. radical equations) involving these skills, but ultimately solvable by factoring/quadratic formula (no complex roots) Evet A: Fuctios ad Algebraic Maipulatio Factorig Square of a sum: ( a + b) = a + ab + b Square of a differece: ( a b) = a ab + b Differece of squares: a b = ( a b )(a + b ) Differece of cubes: a 3 b 3

More information

Essential Question How can you recognize an arithmetic sequence from its graph?

Essential Question How can you recognize an arithmetic sequence from its graph? . Aalyzig Arithmetic Sequeces ad Series COMMON CORE Learig Stadards HSF-IF.A.3 HSF-BF.A. HSF-LE.A. Essetial Questio How ca you recogize a arithmetic sequece from its graph? I a arithmetic sequece, the

More information

Solution: APPM 1360 Final Spring 2013

Solution: APPM 1360 Final Spring 2013 APPM 36 Fial Sprig 3. For this proble let the regio R be the regio eclosed by the curve y l( ) ad the lies, y, ad y. (a) (6 pts) Fid the area of the regio R. (b) (6 pts) Suppose the regio R is revolved

More information

Formula List for College Algebra Sullivan 10 th ed. DO NOT WRITE ON THIS COPY.

Formula List for College Algebra Sullivan 10 th ed. DO NOT WRITE ON THIS COPY. Forula List for College Algera Sulliva 10 th ed. DO NOT WRITE ON THIS COPY. Itercepts: Lear how to fid the x ad y itercepts. Syetry: Lear how test for syetry with respect to the x-axis, y-axis ad origi.

More information

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

More information

Mathematics Extension 2

Mathematics Extension 2 004 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram. Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios

More information

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.

More information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n. ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic

More information

R is a scalar defined as follows:

R is a scalar defined as follows: Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad

More information

CS 70 Second Midterm 7 April NAME (1 pt): SID (1 pt): TA (1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt):

CS 70 Second Midterm 7 April NAME (1 pt): SID (1 pt): TA (1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt): CS 70 Secod Midter 7 April 2011 NAME (1 pt): SID (1 pt): TA (1 pt): Nae of Neighbor to your left (1 pt): Nae of Neighbor to your right (1 pt): Istructios: This is a closed book, closed calculator, closed

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

The Quark Puzzle A 3D printable model and/or paper printable puzzle that allows students to learn the laws of colour charge through inquiry.

The Quark Puzzle A 3D printable model and/or paper printable puzzle that allows students to learn the laws of colour charge through inquiry. The Quark Puzzle A 3D pritable model ad/or paper pritable puzzle that allows studets to lear the laws of colour charge through iquiry. It is available at this lik: https://zeodo.org/record/1252868#.w3ft-gzauk

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information

Orthogonal Function Solution of Differential Equations

Orthogonal Function Solution of Differential Equations Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple,

More information

MA131 - Analysis 1. Workbook 10 Series IV

MA131 - Analysis 1. Workbook 10 Series IV MA131 - Aalysis 1 Workbook 10 Series IV Autum 2004 Cotets 4.19 Rearragemets of Series...................... 1 4.19 Rearragemets of Series If you take ay fiite set of umbers ad rearrage their order, their

More information

Complete Solutions to Supplementary Exercises on Infinite Series

Complete Solutions to Supplementary Exercises on Infinite Series Coplete Solutios to Suppleetary Eercises o Ifiite Series. (a) We eed to fid the su ito partial fractios gives By the cover up rule we have Therefore Let S S A / ad A B B. Covertig the suad / the by usig

More information

(5x 7) is. 63(5x 7) 42(5x 7) 50(5x 7) BUSINESS MATHEMATICS (Three hours and a quarter)

(5x 7) is. 63(5x 7) 42(5x 7) 50(5x 7) BUSINESS MATHEMATICS (Three hours and a quarter) BUSINESS MATHEMATICS (Three hours ad a quarter) (The first 5 miutes of the examiatio are for readig the paper oly. Cadidate must NOT start writig durig this time). ------------------------------------------------------------------------------------------------------------------------

More information

PUTNAM TRAINING PROBABILITY

PUTNAM TRAINING PROBABILITY PUTNAM TRAINING PROBABILITY (Last udated: December, 207) Remark. This is a list of exercises o robability. Miguel A. Lerma Exercises. Prove that the umber of subsets of {, 2,..., } with odd cardiality

More information

A string of not-so-obvious statements about correlation in the data. (This refers to the mechanical calculation of correlation in the data.

A string of not-so-obvious statements about correlation in the data. (This refers to the mechanical calculation of correlation in the data. STAT-UB.003 NOTES for Wedesday 0.MAY.0 We will use the file JulieApartet.tw. We ll give the regressio of Price o SqFt, show residual versus fitted plot, save residuals ad fitted. Give plot of (Resid, Price,

More information

and then substitute this into the second equation to get 5(11 4 y) 3y

and then substitute this into the second equation to get 5(11 4 y) 3y Math E-b Lecture # Notes The priary focus of this week s lecture is a systeatic way of solvig ad uderstadig systes of liear equatios algebraically, geoetrically, ad logically. Eaple #: Solve the syste

More information

Polygons with concurrent medians

Polygons with concurrent medians Polygos with cocurret medias Joh P. Steiberger epartmet of Mathematics, U avis jpsteib@math.ucdavis.ca November 2, 2003 For Thomas achoff o the occasio of his 65th birthday. bstract The medias of a odd-sided

More information

3. One pencil costs 25 cents, and we have 5 pencils, so the cost is 25 5 = 125 cents. 60 =

3. One pencil costs 25 cents, and we have 5 pencils, so the cost is 25 5 = 125 cents. 60 = JHMMC 0 Grade Solutios October, 0. By coutig, there are 7 words i this questio.. + 4 + + 8 + 6 + 6.. Oe pecil costs cets, ad we have pecils, so the cost is cets. 4. A cube has edges.. + + 4 + 0 60 + 0

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

32 estimating the cumulative distribution function

32 estimating the cumulative distribution function 32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio

More information

P.3 Polynomials and Special products

P.3 Polynomials and Special products Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +

More information

Mathematics: Paper 1

Mathematics: Paper 1 GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists

More information

Ray-triangle intersection

Ray-triangle intersection Ray-triagle itersectio ria urless October 2006 I this hadout, we explore the steps eeded to compute the itersectio of a ray with a triagle, ad the to compute the barycetric coordiates of that itersectio.

More information

18.S34 (FALL, 2007) GREATEST INTEGER PROBLEMS. n + n + 1 = 4n + 2.

18.S34 (FALL, 2007) GREATEST INTEGER PROBLEMS. n + n + 1 = 4n + 2. 18.S34 (FALL, 007) GREATEST INTEGER PROBLEMS Note: We use the otatio x for the greatest iteger x, eve if the origial source used the older otatio [x]. 1. (48P) If is a positive iteger, prove that + + 1

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function Summary: Cogrueces j whe divided by, ad determiig the additive order of a iteger mod. As described i the Prelab sectio, cogrueces ca be thought of i terms of coutig with rows, ad for some questios this

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero? 2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a

More information

Primes of the form n 2 + 1

Primes of the form n 2 + 1 Itroductio Ladau s Probles are four robles i Nuber Theory cocerig rie ubers: Goldbach s Cojecture: This cojecture states that every ositive eve iteger greater tha ca be exressed as the su of two (ot ecessarily

More information

6.4 Binomial Coefficients

6.4 Binomial Coefficients 64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter

More information

) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii

) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii square atrix is oe that has the sae uber of rows as colus; that is, a atrix. he idetity atrix (deoted by I, I, or [] I ) is a square atrix with the property that for ay atrix, the product I equals. he

More information

COMP 2804 Solutions Assignment 1

COMP 2804 Solutions Assignment 1 COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig

More information

LESSON 2: SIMPLIFYING RADICALS

LESSON 2: SIMPLIFYING RADICALS High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

7.1 Finding Rational Solutions of Polynomial Equations

7.1 Finding Rational Solutions of Polynomial Equations Name Class Date 7.1 Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Explore Relatig Zeros ad Coefficiets of Polyomial

More information

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe

More information

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio

More information

Calculus 2 Test File Spring Test #1

Calculus 2 Test File Spring Test #1 Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad

More information

Two or more points can be used to describe a rigid body. This will eliminate the need to define rotational coordinates for the body!

Two or more points can be used to describe a rigid body. This will eliminate the need to define rotational coordinates for the body! OINTCOORDINATE FORMULATION Two or more poits ca be used to describe a rigid body. This will elimiate the eed to defie rotatioal coordiates for the body i z r i i, j r j j rimary oits: The coordiates of

More information

ARITHMETIC PROGRESSIONS

ARITHMETIC PROGRESSIONS CHAPTER 5 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first

More information

BITSAT MATHEMATICS PAPER III. For the followig liear programmig problem : miimize z = + y subject to the costraits + y, + y 8, y, 0, the solutio is (0, ) ad (, ) (0, ) ad ( /, ) (0, ) ad (, ) (d) (0, )

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B) SCORING GUIDELINES (Form B) Questio 5 Let f be a fuctio defied o the closed iterval [,7]. The graph of f, cosistig of four lie segmets, is show above. Let g be the fuctio give by g ftdt. (a) Fid g (, )

More information

SAFE HANDS & IIT-ian's PACE EDT-10 (JEE) SOLUTIONS

SAFE HANDS & IIT-ian's PACE EDT-10 (JEE) SOLUTIONS . If their mea positios coicide with each other, maimum separatio will be A. Now from phasor diagram, we ca clearly see the phase differece. SAFE HANDS & IIT-ia's PACE ad Aswer : Optio (4) 5. Aswer : Optio

More information

VICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015

VICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015 VICTORIA JUNIOR COLLEGE Prelimiary Eamiatio MATHEMATICS (Higher ) 70/0 Paper September 05 Additioal Materials: Aswer Paper Graph Paper List of Formulae (MF5) 3 hours READ THESE INSTRUCTIONS FIRST Write

More information

Set 3 Paper 2. Set 3 Paper 2. 1 Pearson Education Asia Limited C

Set 3 Paper 2. Set 3 Paper 2. 1 Pearson Education Asia Limited C . D. A. C. C. C 6. A 7. B 8. D. B 0. A. C. D. B. C. C 6. C 7. C 8. A. D 0. A. D. B. C. A. A 6. D 7. C 8. C. C 0. A. D. D. D. D. A 6. A 7. C 8. B. D 0. D. A. C. D. A. D Sectio A. D ( ) 6. A + a + a a (

More information

INTEGRATION BY PARTS (TABLE METHOD)

INTEGRATION BY PARTS (TABLE METHOD) INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary

More information

Section 6.4: Series. Section 6.4 Series 413

Section 6.4: Series. Section 6.4 Series 413 ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they

More information

Chapter 2. Asymptotic Notation

Chapter 2. Asymptotic Notation Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It

More information

Example: Find the SD of the set {x j } = {2, 4, 5, 8, 5, 11, 7}.

Example: Find the SD of the set {x j } = {2, 4, 5, 8, 5, 11, 7}. 1 (*) If a lot of the data is far from the mea, the may of the (x j x) 2 terms will be quite large, so the mea of these terms will be large ad the SD of the data will be large. (*) I particular, outliers

More information

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL

More information

Representing transformations by matrices

Representing transformations by matrices Teachig Further Mathematics FP Give each pair of studets a copy of the sheet below elarged oto A. Represetig trasformatios by matrices Studets have to multiply the matri by the positio vector of each verte

More information

( ) D) E) NOTA

( ) D) E) NOTA 016 MAΘ Natioal Covetio 1. Which Greek mathematicia do most historias credit with the discovery of coic sectios as a solutio to solvig the Delia problem, also kow as doublig the cube? Eratosthees Meaechmus

More information

Algebra of Least Squares

Algebra of Least Squares October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal

More information

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f, AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

More information

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S )

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Grade 11 Pre-Calculus Mathematics (30S) is desiged for studets who ited to study calculus ad related mathematics as part of post-secodary

More information

Revision Topic 1: Number and algebra

Revision Topic 1: Number and algebra Revisio Topic : Number ad algebra Chapter : Number Differet types of umbers You eed to kow that there are differet types of umbers ad recogise which group a particular umber belogs to: Type of umber Symbol

More information

Mathematics Extension 1

Mathematics Extension 1 016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved

More information

This paper consists of 10 pages with 10 questions. All the necessary working details must be shown.

This paper consists of 10 pages with 10 questions. All the necessary working details must be shown. Mathematics - HG Mar 003 Natioal Paper INSTRUCTIONS.. 3. 4. 5. 6. 7. 8. 9. This paper cosists of 0 pages with 0 questios. A formula sheet is icluded o page 0 i the questio paper. Detach it ad use it to

More information