NONCLASSICAL CONSTRUCTIONS II


 Mae McDaniel
 3 years ago
 Views:
Transcription
1 NONLSSIL ONSTRUTIONS II hristopher Ohrt UL Mthcircle  Nov. 22, 2015 Now we will try ourselves on onceletsteiner constructions. You cn only use n (unmrked) strightedge but you cn ssume tht somewhere in the plne there is one circle (nd its center) given. roblem 1 (onstruction of prllels through given point). (i) Just using n (unmrked) strightedge construct the prllel to bisected line segment (tht is line segment of which you re given the midpoint). (ii) Now ssume you re given circle nd its midpoint. onstruct prllelogrm inscribed in the circle. Hint: Strt with rndom dimeter of the circle nd try using the previous result. (iii) You re still given circle nd its midpoint. onstruct bisected line segment on given line (not necessrily pssing through the circle). Hint: hoose rndom dimeter nd intersect it with the line. Now try using wht you ve lerned so fr. (iv) How does this enble you to give the onceletsteiner construction of the prllel to given line through given point? 1
2 roblem 2 (Reflection on line). Given point nd line, use the sketch to construct its reflection. The second sketch shows you wht you strt out with. Hint: The dimeter contining H is rndomly chosen. The lines HJ nd re prllel. Wht else is prllel? H J 2
3 roblem 3 (Intersection of line nd circle). Use the following sketch to construct the intersection points of the circle round O of rdius O nd the line. The second sketch shows wht you strt out with. The dotted circle cnnot be drwn (with strightedge), so it s imgintively there but you cnnot intersect lines with it directly. Include proof why your construction works. Hint: The point L is freely chosen nywhere on. Things tht look prllel, re most likely prllel. First use ny pprent prellelities nd the intersect theorem to prove your ssertion. Then use the intersect theorem (severl times) to prove tht ll the lines you need re prllel. M L O M O 3
4 roblem 4. Refer to the ppendix for the definition nd bsic properties of the rdicl xis of two circles. Show tht the (three different) rdicl xis of three circles intersect in one point. n you use this to construct the rdicl xis of two circles using stright edge nd compss? 4
5 roblem 5 (Rdicl xis of two circles). Given two circles nd b, use the sketch below to construct their rdicl xis. The second sketch shows wht you strt out with. The dotted circles re given (but not drwn). However, thnks to the previous problem you cn merrily intersect ny line with them. Hint: The point L is chosen freely on. So re the rys contining nd B. Show tht BD is concyclic nd conclude from the previous problem tht therefore R must be on the rdicl xis. n you identify second point on the rdicl xis? During your construction, be mindful tht the only thing you know bout the given circles is their midpoint nd rdius. In prticulr you cn NOT drw them/intersect them with other circles. b L B R D b 5
6 roblem 6. Use the previous two constructions to construct the intersection points of two circles nd thus finish the proof of the onceletsteiner Theorem. 6
7 ppendix: Some Fcts from Geometry Recll the following fcts from geometry. If you re curious you cn sk one of the instructors to give you proofs for them. oncyclic oints In the sketch below the points, B,, nd D re on circle if nd only if the ngles BD nd D gree. Similrly, B,, nd D re on circle if nd only if the ngles B nd ED gree. B D E Rdicl xis The rdicl xis of two circles c 1 nd c 2 is the set of points such tht the tngents through to c 1 nd c 2 hve the sme length. Note tht we cn drw two different tngents through to given circle, but they will lwys hve the sme length. It cn be shown tht the rdicl xis is line perpendiculr to the line connecting the centers of c 1 nd c 2. n you describe the rdicl xis of two intersecting circles? The sketch below shows the rdicl xis of two circles nd point on it such tht the tngentil lengths r 1 nd r 2 gree. r 1 r 2 7
Polar Coordinates. July 30, 2014
Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xyplne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes
More informationTriangles and parallelograms of equal area in an ellipse
1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper
More information9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations
54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show
More informationPolar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a
Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.
More informationGeometric quantities for polar curves
Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions
More informationMATH 118 PROBLEM SET 6
MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the
More informationTranslate and Classify Conic Sections
TEKS 9.6 A.5.A, A.5.B, A.5.D, A.5.E Trnslte nd Clssif Conic Sections Before You grphed nd wrote equtions of conic sections. Now You will trnslte conic sections. Wh? So ou cn model motion, s in E. 49. Ke
More information6.1 Justifying Constructions
Name lass ate 6.1 Justifying onstructions Essential Question: How can you be sure that the result of a construction is valid? Resource Locker Explore 1 Using a Reflective evice to onstruct a erpendicular
More informationSection 10.2 Graphing Polar Equations
Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The
More informationTo use properties of perpendicular bisectors and angle bisectors
52 erpendicular and ngle isectors ontent tandards G.O.9 rove theorems about lines and angles... points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints.
More informationDiffraction and Interference. 6.1 Diffraction. Diffraction grating. Diffraction grating. Question. Use of a diffraction grating in a spectrometer
irction nd Intererence 6.1 irction irction grting Circulr dirction irction nd intererence re similr phenomen. Intererence is the eect o superposition o 2 coherent wves. irction is the superposition o mny
More informationAPPLIED GEOMETRY COORDINATE SYSTEM LINE CONSTRUCTION LINE CONSTRUCTION BISECTING LINE OR ARC LINE CONSTRUCTION
OORDINTE SSTEM PPLIED GEOMETR ( LINE, NGLE, POLGON, R, IRLE, ND UTILITIES) LINE ONSTRUTION 10. 9. 8. 7. 6. 5. 4. 3. 2. Z 1. 0. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. :: 2 steps are used to create one line.
More informationLECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY
LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd
More informationMath Circles Finite Automata Question Sheet 3 (Solutions)
Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how
More informationSpiral Tilings with Ccurves
Spirl Tilings with curves Using ombintorics to Augment Trdition hris K. Plmer 19 North Albny Avenue hicgo, Illinois, 0 chris@shdowfolds.com www.shdowfolds.com Abstrct Spirl tilings used by rtisns through
More information10.4 AREAS AND LENGTHS IN POLAR COORDINATES
65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the
More information23.2 Angle Bisectors of Triangles
Name lass Date 23.2 ngle isectors of Triangles Essential uestion: How can you use angle bisectors to find the point that is equidistant from all the sides of a triangle? Explore Investigating Distance
More informationVector Calculus. 1 Line Integrals
Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q
More informationMath 3 Geogebra Discovery  Equidistance Decemeber 5, 2014
Math 3 Geogebra Discovery  Equidistance Decemeber 5, 2014 Today you and your partner are going to explore two theorems: The Equidistance Theorem and the Perpendicular Bisector Characterization Theorem.
More informationREVIEW, pages
REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute:
More informationFirst Round Solutions Grades 4, 5, and 6
First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,
More informationIntroduction to Planimetry of QuasiElliptic Plane
Originl scientific pper ccepted 4. 11. 2016.. Sliepčević, I. Božić Drgun: Introduction to Plnimetry of QusiElliptic Plne N SLIEPČEVIĆ IVN BOŽIĆ DRGUN Introduction to Plnimetry of QusiElliptic Plne Introduction
More informationExample. Check that the Jacobian of the transformation to spherical coordinates is
lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.
More informationCS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates
Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the
More informationUnit 6 Lesson 1 Circle Geometry Properties Project
Unit 6 Lesson 1 Circle Geometry Properties Project Name Part A Look up and define the following vocabulary words. Use an illustration where appropriate. Some of this vocabulary can be found in the glossary
More informationSection 16.3 Double Integrals over General Regions
Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is
More informationUNIT 3 CIRCLES AND VOLUME Lesson 3: Constructing Tangent Lines Instruction
Prerequisite Skills This lesson requires the use of the following skills: understanding the relationship between perpendicular lines using a compass and a straightedge constructing a perpendicular bisector
More informationSOLVING TRIANGLES USING THE SINE AND COSINE RULES
Mthemtics Revision Guides  Solving Generl Tringles  Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES
More informationNotes on Spherical Triangles
Notes on Spheril Tringles In order to undertke lultions on the elestil sphere, whether for the purposes of stronomy, nvigtion or designing sundils, some understnding of spheril tringles is essentil. The
More informationc The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?
3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem
More informationAlgorithms for Memory Hierarchies Lecture 14
Algorithms for emory Hierrchies Lecture 4 Lecturer: Nodri Sitchinv Scribe: ichel Hmnn Prllelism nd Cche Obliviousness The combintion of prllelism nd cche obliviousness is n ongoing topic of reserch, in
More informationDetermine currents I 1 to I 3 in the circuit of Fig. P2.14. Solution: For the loop containing the 18V source, I 1 = 0.
Prolem.14 Determine currents 1 to 3 in the circuit of Fig. P.14. 1 A 18 V Ω 3 A 1 8 Ω 1 Ω 7 Ω 4 Ω 3 Figure P.14: Circuit for Prolem.14. For the loop contining the 18V source, Hence, 1 = 1.5 A. KCL t node
More informationSlopes of Lines Notes What is slope?
Slopes of Lines Notes What is slope? Find the slope of each line. 1 Find the slope of each line. Find the slope of the line containing the given points. 6, 2!!"#! 3, 5 4, 2!!"#! 4, 3 Find the slope of
More informationS. Stirling Page 1 of 14
3.1 Duplicating Segments and ngles [and riangles] hese notes replace pages 144 146 in the book. You can read these pages for extra clarifications. Instructions for making geometric figures: You can sketch
More informationSection 6.1 Law of Sines. Notes. Oblique Triangles  triangles that have no right angles. A c. A is acute. A is obtuse
Setion 6.1 Lw of Sines Notes. Olique Tringles  tringles tht hve no right ngles h is ute h is otuse Lw of Sines  If is tringle with sides,, nd, then sin = sin = sin or sin = sin = sin The miguous se (SS)
More informationINTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS
CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite
More informationDataflow Language Model. DataFlow Models. Applications of Dataflow. Dataflow Languages. Kahn process networks. A Kahn Process (1)
The slides contin revisited mterils from: Peter Mrwedel, TU Dortmund Lothr Thiele, ETH Zurich Frnk Vhid, University of liforni, Riverside Dtflow Lnguge Model Drsticlly different wy of looking t computtion:
More informationExercise 11. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION
Exercise 11 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You
More informationL7 Constructions 7.1 Construction Introduction Per Date
7.1 Construction Introduction Per Date In pairs, discuss the meanings of the following vocabulary terms. The first two you should attempt to recall from memory, and for the rest you should try to agree
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Prctice Skills Prctice for Lesson.1 Nme Dte Tiling Bthroom Wll Simplifying Squre Root Expressions Vocbulry Mtch ech definition to its corresponding term. 1. n expression tht involves root. rdicnd
More informationUp and Down.  Circle Theorems 2: The Converse of the Chord Theorem 
 Circle Theorems 2: The Converse of the Chord Theorem  Revision Label the circle diagram showing: the circumference the centre a diameter a chord a radius State the Chord Theorem. Checkpoint An Example
More informationStudy Guide: Similarity and Dilations
Study Guide: Similarity and ilations ilations dilation is a transformation that moves a point a specific distance from a center of dilation as determined by the scale factor (r). Properties of ilations
More information(CATALYST GROUP) B"sic Electric"l Engineering
(CATALYST GROUP) B"sic Electric"l Engineering 1. Kirchhoff s current l"w st"tes th"t (") net current flow "t the junction is positive (b) Hebr"ic sum of the currents meeting "t the junction is zero (c)
More informationTangents and Chords Off On a Tangent
Tangents and Chords SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Create Representations, Quickwrite CTIVITY 4.1 circle is
More information(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ
Tringles nd Trigonometry Prepred y: S diyy Hendrikson Nme: Dte: Suppose we were sked to solve the following tringles: Notie tht eh tringle hs missing informtion, whih inludes side lengths nd ngles. When
More informationb = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 )
***************** Disclimer ***************** This represents very brief outline of most of the topics covered MA261 *************************************************** I. Vectors, Lines nd Plnes 1. Vector
More information6.1 Warm Up The diagram includes a pair of congruent triangles. Use the congruent triangles to find the value of x in the diagram.
6.1 Warm Up The diagram includes a pair of congruent triangles. Use the congruent triangles to find the value of x in the diagram. 1. 2. Write a proof. 3. Given: P is the midpoint of MN and TQ. Prove:
More informationGeometric Constructions
Geometric onstructions (1) opying a segment (a) Using your compass, place the pointer at Point and extend it until reaches Point. Your compass now has the measure of. (b) Place your pointer at, and then
More informationArduino for Model Railroaders
Steve Mssikker Arduino for Model Rilroders Ornge Book Protocol 2 Full Description November 28 Tble of contents Dontors Documenttion Kit V.48 Pge 2 I wnt to tke the time to sincerely thnk you for your
More informationSTRAND H: Angle Geometry
Mathematics SKE, Strand H UNIT H3 onstructions and Loci: Text STRND H: ngle Geometry H3 onstructions and Loci Text ontents Section H3.1 Drawing and Symmetry H3.2 onstructing Triangles and ther Shapes H3.3
More informationUniversity of North CarolinaCharlotte Department of Electrical and Computer Engineering ECGR 4143/5195 Electrical Machinery Fall 2009
Problem 1: Using DC Mchine University o North CrolinChrlotte Deprtment o Electricl nd Computer Engineering ECGR 4143/5195 Electricl Mchinery Fll 2009 Problem Set 4 Due: Thursdy October 8 Suggested Reding:
More information5.1. Perpendiculars and Bisectors. What you should learn
age 1 of 8 5.1 erpendiculars and isectors What you should learn GOL 1 Use properties of perpendicular bisectors. GOL 2 Use properties of angle bisectors to identify equal distances, such as the lengths
More information1. Construct the perpendicular bisector of a line segment. Or, construct the midpoint of a line segment. 1. Begin with line segment XY.
1. onstruct the perpendicular bisector of a line segment. Or, construct the midpoint of a line segment. 1. egin with line segment. 2. lace the compass at point. djust the compass radius so that it is more
More informationChallenges from Ancient Greece
Challenges from ncient Greece Mathematical goals Make formal geometric constructions with a variety of tools and methods. Use congruent triangles to justify geometric constructions. Common Core State Standards
More informationSpherical Geometry. This is an article from my home page:
Spheril Geometry This is n rtile from my home pge: www.olewitthnsen.dk Ole WittHnsen nov. 6 Contents. Geometry on sphere.... Spheril tringles...3. Polr tringles...4 3. The rightngle spheril tringle...6
More informationKirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):
SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween
More informationProbability and Statistics P(A) Mathletics Instant Workbooks. Copyright
Proility nd Sttistis Student Book  Series K P(A) Mthletis Instnt Workooks Copyright Student Book  Series K Contents Topis Topi  Review of simple proility Topi  Tree digrms Topi  Proility trees Topi
More informationStudy Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms
Study Guide # 1 MA 26100  Fll 2018 1. Vectors in R 2 nd R 3 () v =, b, c = i + b j + c k; vector ddition nd subtrction geometriclly using prllelogrms spnned by u nd v; length or mgnitude of v =, b, c,
More information9.1 and 9.2 Introduction to Circles
Date: Secondary Math 2 Vocabulary 9.1 and 9.2 Introduction to Circles Define the following terms and identify them on the circle: Circle: The set of all points in a plane that are equidistant from a given
More informationMaterials: Computer lab or set of calculators equipped with Cabri Geometry II and lab worksheet.
Constructing Perpendiculars Lesson Summary: Students will complete the basic compass and straight edge constructions commonly taught in first year high school Geometry. Key Words: perpendicular, compass,
More informationGeometry Ch 3 Vertical Angles, Linear Pairs, Perpendicular/Parallel Lines 29 Nov 2017
3.1 Number Operations and Equality Algebraic Postulates of Equality: Reflexive Property: a=a (Any number is equal to itself.) Substitution Property: If a=b, then a can be substituted for b in any expression.
More informationUnit 6 Quadrilaterals
Unit 6 Quadrilaterals ay lasswork ay Homework Monday Properties of a Parallelogram 1 HW 6.1 11/13 Tuesday 11/14 Proving a Parallelogram 2 HW 6.2 Wednesday 11/15 Thursday 11/16 Friday 11/17 Monday 11/20
More informationStudent Book SERIES. Patterns and Algebra. Name
E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing
More informationUnit 6: Quadrilaterals
Name: Period: Unit 6: Quadrilaterals Geometry Honors Homework Section 6.1: Classifying Quadrilaterals State whether each statement is true or false. Justify your response. 1. All squares are rectangles.
More informationMeasuring and Drawing Angles and Triangles
NME DTE Measuring and Drawing ngles and Triangles Measuring an angle 30 arm origin base line 0 180 0 If the arms are too short to reach the protractor scale, lengthen them. Step 1: lace the origin of the
More information+ sin bsin. sin. tan
6. Spheril rignmetri Frmule Just s in plne gemetry, there re useful trignmetri frmule whih relte the sides nd vertex ngles f spheril tringles: Csine Frmul [6.1] s ss + sin sin s Sine Frmul [6.] sin sin
More informationGeometry Unit 3 Note Sheets Date Name of Lesson. Slopes of Lines. Partitioning a Segment. Equations of Lines. Quiz
Date Name of Lesson Slopes of Lines Partitioning a Segment Equations of Lines Quiz Introduction to Parallel and Perpendicular Lines Slopes and Parallel Lines Slopes and Perpendicular Lines Perpendicular
More informationITEC2620 Introduction to Data Structures
/5/20 ITEC220 Introdution to Dt Strutures Leture 0 Gme Trees TwoPlyer Gmes Rules for gme define the sttespe Nodes re gme sttes Links re possile moves Build serh tree y rute fore Exmple I Exmple II A Our
More information16.1 Segment Length and Midpoints
Name lass ate 16.1 Segment Length and Midpoints Essential Question: How do you draw a segment and measure its length? Explore Exploring asic Geometric Terms In geometry, some of the names of figures and
More informationName Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines
Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines Two lines are if they are coplanar and do not intersect. Skew lines. Two
More informationThe Formal Proof of a Theorem
.7 The Formal Proof of a Theorem 5.7 The Formal Proof of a Theorem KY ONPTS Formal Proof of a Theorem onverse of a Theorem Picture Proof (Informal) of a Theorem Recall from Section. that statements that
More informationSection 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.
Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line
More informationName No. Geometry 93 1) Complete the table: Name No. Geometry 91 1) Name a secant. Name a diameter. Name a tangent. Name No. Geometry 92 1) Find JK
Geometry 91 1) Name a secant 1) Complete the table: Name a diameter Name a tangent Geometry 92 1) Find JK 2) Find the measure of 1 Geometry 92 2) 3) At 2:00 the hands of a clock form an angle of 2)
More informationMONOCHRONICLE STRAIGHT
UPDATED 092010 HYDROCARBON Hydrocrbon is ponchostyle cowl in bulkyweight yrn, worked in the round. It ws designed to be s prcticl s it is stylish, with shping tht covers the neck nd shoulders nd the
More informationUsing Geometry. 9.1 Earth Measure. 9.2 Angles and More Angles. 9.3 Special Angles. Introduction to Geometry and Geometric Constructions...
Using Geometry Recognize these tools? The one on the right is a protractor, which has been used since ancient times to measure angles. The one on the left is a compass, used to create arcs and circles.
More informationLesson 3.1 Duplicating Segments and Angles
Lesson 3.1 Duplicating Segments and ngles Name eriod Date In Exercises 1 3, use the segments and angles below. omplete the constructions on a separate piece of paper. S 1. Using only a compass and straightedge,
More informationUniversity of Houston High School Mathematics Contest Geometry Exam Spring 2016
University of Houston High School Mathematics ontest Geometry Exam Spring 016 nswer the following. Note that diagrams may not be drawn to scale. 1. In the figure below, E, =, = 4 and E = 0. Find the length
More informationUNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 3: Constructing Polygons Instruction
rerequisite Skills This lesson requires the use of the following skills: using a compass copying and bisecting line segments constructing perpendicular lines constructing circles of a given radius Introduction
More informationIndicate whether the statement is true or false.
MATH 121 SPRING 2017  PRACTICE FINAL EXAM Indicate whether the statement is true or false. 1. Given that point P is the midpoint of both and, it follows that. 2. If, then. 3. In a circle (or congruent
More informationSTUDY GUIDE, CALCULUS III, 2017 SPRING
TUY GUIE, ALULU III, 2017 PING ontents hpter 13. Functions of severl vribles 1 13.1. Plnes nd surfces 2 13.2. Grphs nd level curves 2 13.3. Limit of function of two vribles 2 13.4. Prtil derivtives 2 13.5.
More informationSketchUp Project Gear by Mark Slagle
SketchUp Project Gear by Mark Slagle This lesson was donated by Mark Slagle and is to be used free for education. For this Lesson, we are going to produce a gear in SketchUp. The project is pretty easy
More informationAxiom A1: To every angle there corresponds a unique, real number, 0 < < 180.
Axiom A1: To every angle there corresponds a unique, real number, 0 < < 180. We denote the measure of ABC by m ABC. (Temporary Definition): A point D lies in the interior of ABC iff there exists a segment
More informationLecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is
More informationTheme: Don t get mad. Learn mod.
FERURY When 1 is divided by 5, the reminder is. nother wy to sy this is opyright 015 The Ntionl ouncil of Techers of Mthemtics, Inc. www.nctm.org. ll rights reserved. This mteril my not be copied or distributed
More informationEfficient and Resilient Key Discovery based on PseudoRandom Key PreDeployment
Efficient nd Resilient Key Discovery sed on PseudoRndom Key PreDeployment p. 1 Efficient nd Resilient Key Discovery sed on PseudoRndom Key PreDeployment Roerto Di Pietro, Luigi V. Mncini, nd Alessndro
More informationChapter 5. Drawing a cube. 5.1 One and twopoint perspective. Math 4520, Spring 2015
Chapter 5 Drawing a cube Math 4520, Spring 2015 5.1 One and twopoint perspective In Chapter 5 we saw how to calculate the center of vision and the viewing distance for a square in one or twopoint perspective.
More informationECE 274 Digital Logic Fall 2009 Digital Design
igitl Logic ll igitl esign MW :PM, IL Romn Lysecky, rlysecky@ece.rizon.edu http://www.ece.rizon.edu/~ece hpter : Introduction Slides to ccompny the textbook igitl esign, irst dition, by rnk Vhid, John
More informationTIME: 1 hour 30 minutes
UNIVERSITY OF AKRON DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING 4400: 34 INTRODUCTION TO COMMUNICATION SYSTEMS  Spring 07 SAMPLE FINAL EXAM TIME: hour 30 minutes INSTRUCTIONS: () Write your nme
More informationMisty. Sudnow Dot Songs
Sudnow Dot Songs isty T The Dot Song is nottionl system tht depicts voiced chords in wy where the nonmusic reder cn find these firly redily. But the Dot Song is not intended be red, not s sight reder
More informationMAYWOOD. Hospitality & ConferenCe tables FURNITURE CORP. GSA Approved! Contract #GS28F0050W. Established 1918
MAYWOOD FURNITURE CORP. Mnufctured in the U.S.A. Estblished 1918 Hospitlity & ConferenCe tbles www.mywood.com GSA Approved! Contrct #GS28F0050W bout us Circ 1930 Mywood Furniture Corportion ws estblished
More informationFubini for continuous functions over intervals
Fuini for ontinuous funtions over intervls We first prove the following theorem for ontinuous funtions. Theorem. Let f(x) e ontinuous on ompt intervl =[, [,. Then [, [, [ [ f(x, y)(x, y) = f(x, y)y x =
More informationFolding Activity 3. Compass Colored paper Tape or glue stick
Folding Activity 3 Part 1 You re not done until everyone in your group is done! If you finish before someone else, help them finish before starting on the next part. You ll need: Patty paper Ruler Sharpie
More informationUnit 6 Guided Notes. Task: To discover the relationship between the length of the midsegment and the length of the third side of the triangle.
Unit 6 Guided Notes Geometry Name: Period: Task: To discover the relationship between the length of the midsegment and the length of the third side of the triangle. Materials: This paper, compass, ruler
More informationModule 9. DC Machines. Version 2 EE IIT, Kharagpur
Module 9 DC Mchines Version EE IIT, Khrgpur esson 40 osses, Efficiency nd Testing of D.C. Mchines Version EE IIT, Khrgpur Contents 40 osses, efficiency nd testing of D.C. mchines (esson40) 4 40.1 Gols
More informationStudent Book SERIES. Fractions. Name
D Student Book Nme Series D Contents Topic Introducing frctions (pp. ) modelling frctions frctions of collection compring nd ordering frctions frction ingo pply Dte completed / / / / / / / / Topic Types
More informationArcs, Central and Inscribed Angles Coming Full Circle
rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: Shared Reading, Summarize/Paraphrase/Retell, Visualization Chris loves to play soccer. When he was seven years old, his family enrolled him
More informationMEASURE THE CHARACTERISTIC CURVES RELEVANT TO AN NPN TRANSISTOR
Electricity Electronics Bipolr Trnsistors MEASURE THE HARATERISTI URVES RELEVANT TO AN NPN TRANSISTOR Mesure the input chrcteristic, i.e. the bse current IB s function of the bse emitter voltge UBE. Mesure
More informationChapter 12 Vectors and the Geometry of Space 12.1 Threedimensional Coordinate systems
hpter 12 Vectors nd the Geometry of Spce 12.1 Threedimensionl oordinte systems A. Three dimensionl Rectngulr oordinte Sydstem: The rtesin product where (x, y, z) isclled ordered triple. B. istnce: R 3
More informationConstruction Junction, What s your Function?
Construction Junction, What s your Function? Brian Shay Teacher and Department Chair Canyon Crest Academy Brian.Shay@sduhsd.net @MrBrianShay Session Goals Familiarize ourselves with CCSS and the GSE Geometry
More informationTopic 20: Huffman Coding
Topic 0: Huffmn Coding The uthor should gze t Noh, nd... lern, s they did in the Ark, to crowd gret del of mtter into very smll compss. Sydney Smith, dinburgh Review Agend ncoding Compression Huffmn Coding
More information