Geom 3eTE.1101.X 598-604 - Portal.mywccc
11-1Space Figures andCross Sections11-11. PlanObjectives12To recognize polyhedra andtheir partsTo visualize cross sections ofspace figuresExamples12345Identifying Vertices, Edges,and FacesUsing Euler’s FormulaVerifying Euler’s FormulaDescribing a Cross SectionDrawing a Cross SectionGO for HelpWhat You’ll LearnCheck Skills You’ll Need To recognize polyhedra andFor each exercise, make a copy of the cubeat the right. Shade the plane that containsthe indicated points. 1–7. See back of book.their parts To visualize cross sections ofspace figures. . . And WhyTo learn about medicaltechniques, as in Exercise 44.1. A, B, and C2. A, B, and G3. A, C, and G4. A, D, and G5. F, D, and G6. B, D, and GLesson 1-3BCDAF7. the midpoints of AD, CD, EH, and GHGEHNew Vocabulary polyhedron face edge vertex cross sectionMath BackgroundThe references to plane figuresare somewhat informal. Explain,for example, that a base of acylinder is not technically a circle;it is a circle together with thecircle’s interior. Similarly, a faceof a polyhedron is not actually apolygon; it is a polygon togetherwith its interior.1Identifying Parts of a PolyhedronA polyhedron is a three-dimensionalfigure whose surfaces are polygons. Eachpolygon is a face of the polyhedron.An edge is a segment that is formed by theintersection of two faces. A vertex is apoint where three or more edges intersect.Vocabulary TipPolyhedron comes fromthe Greek poly for“many” and hedron for“side.” A cube is apolyhedron with six sides,or faces, each of which isa square.More Math Background: p. 596C1EXAMPLEFacesEdgeVertexIdentifying Vertices, Edges, and Facesa. How many vertices are there in the polyhedronat the right? List them.Lesson Planning andResourcesHThere are five vertices: D, E, F, G, and H.See p. 596E for a list of theresources that support this lesson.There are eight edges: DE , EF, FG, GD,DH, EH, FH, and GH.PowerPointFGb. How many edges are there? List them.DEc. How many faces are there? List them.Bell Ringer PracticeThere are five faces: #DEH, #EFH, #FGH, #GDH, and the quadrilateralDEFG.RCheck Skills You’ll NeedFor intervention, direct students to:Quick CheckIdentifying PlanesLesson 1-3: Example 4Extra Skills, Word Problems, ProofPractice, Ch. 11 List the vertices, edges, and faces of the polyhedron.R, S, T, U, V; RS, RU, RT , VS, VU, VT , SU,UT , TS; kRSU, kRUT, kRTS, kVSU,kVUT, kVTSSTUV598Chapter 11 Surface Area and VolumeSpecial NeedsBelow LevelL1Review nets by having students cut-out various netsand form their corresponding three-dimensionalfigures. Clarify that there are many possible nets forthe same polyhedron.598learning style: tactileL2Some students may think that spheres and cylindersare polyhedrons. Emphasize that the surfaces ofpolyhedrons are polygons, whose sides must be linesegments. Have students draw examples of polygonson the board.learning style: visual
2. TeachLeonhard Euler, a Swiss mathematician, discovered a relationship among thenumbers of faces, vertices, and edges of any polyhedron. The result is known asEuler’s Formula.Key ConceptsFormulaGuided InstructionEuler’s FormulaThe numbers of faces (F), vertices (V), and edges (E) of a polyhedron arerelated by the formula F V E 2.2EXAMPLE1Using Euler’s FormulaPowerPointAdditional Examples1 How many vertices, edges, andfaces of the polyhedron are there?List them.The polyhedron has 2 hexagons and 6 rectanglesfor a total of 8 faces.The 2 hexagons have a total of 12 edges.The 6 rectangles have a total of 24 edges.If the hexagons and rectangles are joined to form apolyhedron, each edge is shared by two faces. Therefore, thenumber of edges in the polyhedron is one half of the total of 36, or 18.ConnectionEuler’s Formula8 V 18 2Substitute.V 12Euler’s Formula applies to thepolyhedron suggested by thepanels on a volleyball.Quick CheckF V E 2AEJDICGH10 vertices, 15 edges, and 7faces; A, B, C, D, E, F, G, H, I, J;AF, BG, CH, DI, EJ, AB, BC, CD,DE, EA, FG, GH, HI, IJ, JF; ;pentagons ABCD and FGHIJ, andquadrilaterals ABGF, BCHG,CDIH, and EAFJ2 Use Euler’s Formula to find the number of edges on a polyhedron witheight triangular faces. 12 edgesIn two dimensions, Euler’s Formula reduces toF V E 1where F is the number of regions formed by V vertices linked by E segments.EXAMPLEBFSimplify.Count the number of vertices in the figure to verify the result.3Teaching TipEncourage students to worksystematically as they list thevertices, edges, and faces.Count faces and edges. Then use Euler’s Formula to findthe number of vertices in the polyhedron at the right.Real-WorldEXAMPLE2 Use Euler’s Formula to find thenumber of edges of a polyhedronwith 6 faces and 8 vertices.12 edgesVerifying Euler’s FormulaVerify Euler’s Formula for a two-dimensionalnet of the solid in Example 2.3 Using the pentagonal prism inAdditional Example 1, verifyEuler’s Formula. Then draw a netfor the figure and verify Euler’sFormula for the two-dimensionalfigure. 7 10 15 2Draw a net:Count the regions: F 8Count the vertices: V 22Count the segments: E 298 22 29 1Quick Check3 The figure at the right is a trapezoidal prism.a. Verify Euler’s formula F V E 2 forthe prism. 6 8 12 2b. Draw a net for the prism. See margin.c. Verify Euler’s formula F V E 1 foryour two-dimensional net.Sample: 6 14 19 17 18 24 1Quick CheckLesson 11-1 Space Figures and Cross SectionsAdvanced Learners5993.English Language Learners ELLL4Have students determine if values of F, V, and E thatsatisfy Euler’s Formula make an existing polyhedron.learning style: verbalMake sure students understand the differencebetween polyhedron and polygon. Show models ofdifferent polyhedrons and cutouts of differentpolygons.learning style: visual599
Guided Instruction21Describing Cross SectionsError Prevention!Students may think the plane ofa cross section must be horizontalor vertical. Show a cross sectionof an apple or orange cut alonga plane that is neither horizontalnor vertical.5EXAMPLEA cross section is the intersection of a solidand a plane. You can think of a cross sectionas a very thin slice of the selsPupil4Teaching TipPoint out that the exampleassumes that the bottom face ofthe cube is horizontal. Ask: If thecube were tilted slightly, whatmight the cross section look like?Sample: parallelogramReal-WorldConnectionEXAMPLEDescribing a Cross SectionDescribe each cross section.a.Cross sections are used tostudy the anatomy of the eye.b.Visual LearnersThe cross section is a square.Encourage students to slice cubesof butter, ice cream, or modelingclay at home to investigate howplanes may intersect cubes.Quick Check4. Size of sketchesmay vary, Samples:PowerPointAdditional ExamplesThe cross section is a triangle.4 For the funnel shown, sketch each of the following.a. a horizontal cross section a–b. See left.b. a vertical cross section that contains the axis ofsymmetrya.To draw a cross section, you can sometimes use the ideafrom Postulate 1-3 that the intersection of two planes is exactly one line.4 Describe this cross section.b.5EXAMPLEDrawing a Cross SectionVisualization Draw and describe a crosssection formed by a vertical plane intersectingthe front and right faces of the cube.triangle5 Draw and describe a crosssection formed by a vertical planeintersecting the top and bottomfaces of a cube. Check students’work; square or rectangle.Resources Daily Notetaking Guide 11-1A vertical plane cuts the vertical facesof the cube in parallel segments.Draw the parallel segments.5. squareJoin their endpoints. Shade the cross section.L3 Daily Notetaking Guide 11-1—L1Adapted InstructionClosureWhat is a polyhedron and how isEuler’s Formula related to it?A polyhedron is a threedimensional figure whosesurfaces are polygons. Euler’sFormula relates the number offaces (F), vertices (V), and edges(E) of a polyhedron such thatF V E 2.600The cross section is a rectangle.Quick Check6005 Draw and describe the cross section formed by a horizontal plane intersecting theleft and right faces of the cube. See left.Chapter 11 Surface Area and Volume17. rectangle18. square19. rectangle
EXERCISESFor more exercises, see Extra Skill, Word Problem, and Proof Practice.3. PracticePractice and Problem SolvingAssignment GuidePractice by ExampleExample 1For each polyhedron, how many vertices, edges, and faces are there? List them.M1.2.4, 6, 4(page 598)P(page 599)UPDVFEO3.CA1–3. See back ofbook for lists.NExample 2BQGHS8, 12, 6RXWY(page 599)Example 4(page 600)11.56-6061-68To check students’ understandingof key skills and concepts, go overExercises 8, 16, 22, 30, 32.6. Faces: 20 12Edges: 30Vertices: jExercises 13–15 As a class,explore how the shape of thecross section changes with theorientation of the plane.Verify Euler’s Formula for each polyhedron. Then draw a net for the figure andverify Euler’s Formula for the two-dimensional figure. 10–12. See back of book.10.13-19, 21-26, 39-45C Challenge46-55Homework Quick Check5. Faces: 8 12Edges: jVertices: 6Use Euler’s Formula to find the number of vertices in each polyhedrondescribed below.597. 6 square faces8. 5 faces: 1 rectangle9. 9 faces: 1 octagon8and 4 trianglesand 8 trianglesExample 32 A BTest PrepMixed ReviewT10, 15, 7Use Euler’s Formula to find the missing number.4. Faces: j8Edges: 15Vertices: 91 A B 1-12, 20, 27-3812.Exercise 19 Draw a cube on theboard, using a different colorfor each set of opposite edges.Point out that opposite edgesare parallel and are not on thesame face.Describe each cross section. 13. two concentric circles13.14.15.rectangleGPS Guided Problem SolvingL3L4Enrichmenttriangle16. For the nut shown, sketch each of following.a. a horizontal cross section a–b. See back of book.b. a vertical cross section that contains the verticalline of symmetryL2ReteachingL1Adapted PracticePracticeNameClassL3DatePractice 10-1Space Figures and Nets1. Choose the nets that will fold to make a cube.A.Example 5(page 600)Visualization Draw and describe a cross section formedby a vertical plane intersecting the cube as follows.17–19. See margin.17. The vertical plane intersects the front and left faces of the cube.B.C.D.Draw a net for each figure. Label each net with its appropriate dimensions.2.16 cm7 cm2 cm3.8 cm4.1 cm2 cm32 cm1 cm40 cmMatch each three-dimensional figure with its net.5.6.7.8.18. The vertical plane intersects opposite faces of the cube.A.19. The vertical plane contains opposite edges of the cube.B.C.D.9. Choose the nets that will fold to make a pyramid with a square base.A.Lesson 11-1 Space Figures and Cross SectionsB.C.D.601 Pearson Education, Inc. All rights reserved.AUse Euler’s Formula to find the missing number.10. Faces: 5Edges: 7Vertices: 511. Faces: 7Edges: 9Vertices: 612. Faces: 8Edges: 18Vertices: 7601
Exercise 38 Ask: What do youknow about a cube that mighthelp you solve this problem?A cube has 6 square faces.Connection to CalculusBApply Your Skills21. rectangleExercises 27–29 Formulas for thevolumes of more complicatedsolids of revolution are developedin calculus.20. a. Open-Ended Sketch a polyhedron whose faces are all rectangles. Label thelengths of its edges. a–b. See back of book.b. Use graph paper to draw two different nets for the polyhedron.Visualization Draw and describe a cross section formedby a plane intersecting the cube as follows. 21–23. See left.21. The plane is tilted and intersects the left and right facesof the cube.22. The plane contains opposite horizontal edges of the cube.Connection to Astronomy22. rectangle23. The plane cuts off a corner of the cube.Exercise 30 Early scientists usedDescribe the cross section shown.trianglecircle24.25.Platonic solids to attempt toexplain the universe. Havestudents investigate some ofthese explanations.26.2 trapezoids23. triangleVisualization A plane region that revolves completely about a line sweeps out asolid of revolution. Use the sample to help you describe the solid of revolution youget by revolving each region about line .Sample: Revolve therectangular region about theline / and you get a cylinderas a solid of revolution.27.ᐉᐉ28.ᐉᐉcylinder attached to a cone29.ᐉsphereconeSports Equipment Some balls are made from panels thatsuggest polygons. The ball then suggests a polyhedronto which Euler’s Formula, F V E 2, applies.30. A soccer ball suggests a polyhedron with 20 regularhexagons and 12 regular pentagons. How manyvertices does this polyhedron have? 6031. Show how Euler’s Formula applies to the polyhedronsuggested by the volleyball pictured on page 599. (Hint: It has 6 sets of 3 panels.)18 32 48 2Euler’s Formula F V E 1 applies to any two-dimensional network where Fis the number of regions formed by V vertices linked by E edges (or paths). VerifyEuler’s Formula for each network shown.46.GPS 32.47.GO48.49.60250.34.nlineHomework HelpVisit: PHSchool.comWeb Code: aue-110160233.6 4 9 14 6 9 135. Draw a network of your own. Verify Euler’s Formula for it.Check students’ work.Chapter 11 Surface Area and Volume51.52.53.54.5 5 9 1
36. There are five regular polyhedrons. They are called regular because all theirfaces are congruent regular polygons, and the same number of faces meet ateach vertex. They are also called Platonic Solids after the Greek philosopherPlato (427–347 B.C.).4. Assess & ReteachPowerPointLesson Quiz1. Draw a net for the nnectionA fluorite crystal forms as aregular octahedron.IcosahedronSample:Dodecahedrona. Match each net below with a Platonic Solid.A.B.C.D.E.A. icosahedronB. octahedronC. tetrahedronD. hexahedron36b. regular triangular pyramid, cube E. dodecahedronb. The first two Platonic solids have more familiar names. What are they?c. Verify that Euler’s Formula is true for the first three Platonic solids.4 4 6 2, 6 8 12 2, 8 6 12 237. Multiple Choice A cube has a net with area 216 in.2. How long is an edge ofthe cube? A6 in.15 in.36 in.54 in.Use Euler’s Formula to solve.2. A polyhedron with 12 verticesand 30 edges has how manyfaces? 203. A polyhedron with 2 octagonalfaces and 8 rectangular faceshas how many vertices? 164. Describe the cross section.Draw each object. Then draw a horizontal and a vertical cross section.38. a golf tee39. a football40. a baseball bat41. a banana42. a pear43. a bagel38–43. Check students’ work.44. Writing Cross sections are used in medical training and research. Research andwrite a paragraph on how magnetic resonance imaging (MRI) is used to studycross sections of the brain. Check students’ work.CChallenge45. Draw a solid that has the following cross sections.45.circle5. Draw and describe a crosssection formed by a verticalplane cutting the left and backfaces of a cube. Checkstudents’ drawings; rectangle.Alternative AssessmenthorizontalverticalVisualization Draw a plane intersecting a cube to get the cross section indicated.46. scalene triangle47. isosceles triangle48. equilateral triangle49. trapezoid50. isosceles trapezoid51. parallelogram52. rhombuslesson quiz, PHSchool.com, Web Code: aua-110153. pentagon46–54. See margin.Have each student bring a realworld polyhedron to class. Havethem verify Euler’s Formula andthen draw a net for the solid.54. hexagonLesson 11-1 Space Figures and Cross Sections603603
Test PrepTest PrepResourcesMultiple ChoiceFor additional practice with avariety of test item formats: Standardized Test Prep, p. 657 Test-Taking Strategies, p. 652 Test-Taking Strategies withTransparenciesFor Exercises 55–56, you may need Euler’s Formula, F V E 2.55. A polyhedron has four vertices and six edges. How many faces does it have?BA. 2B. 4C. 5D. 1056. A polyhedron has three rectangular faces and two triangular faces.How many vertices does it have? GF. 5G. 6H. 10J. 1257. The plane is horizontal. What best describesthe shape of the cross section? DA. rhombusB. trapezoidC. parallelogramD. square58. The plane is vertical. What best describesthe shape of the cross section? JF. pentagonG. squareH. rectangleJ. triangle59. [2] a. squareShort Responseb. Answers may vary.Sample: trapezoid59. Draw and describe a cross section formedby a plane intersecting a cube as follows.a. The plane is parallel to a horizontal face of the cube.b. The plane cuts off two corners of the cube.a–b. See margin.Mixed Review[1] only 1 correctdrawingGO forHelpLesson 10-860. Probability A shuttle bus to an airport terminal leaves every 20 min from aremote parking lot. Draw a geometric model and find the probability that atraveler who arrives at a random time will have to wait at least 8 min for the60%bus to leave the parking lot.0 4 8 12 16 2061. Games A dartboard is a circle with a 12-in. radius. You throw a dart that hitsthe dartboard. What is the probability that the dart lands within 6 in. of thecenter of the dartboard? 25%Lesson 10-3Lesson 8-3Find the area of each equilateral triangle with the given measure. Leave answers insimplest radical form.62. side 2 ft63. apothem 8 cm192"3 cm2"3 ft2Find the value of x to the nearest tenth.65.4.764. radius 100 in.7500"3 in.266. 8.365 xx636 1067. The lengths of the diagonals of a rhombus are 4 cm and 6 cm. Find themeasures of the angles of the rhombus to the nearest degree. 67 and 113604604Chapter 11 Surface Area and Volume
Faces: j 5. Faces: 8 6. Faces: 20 Edges: 15 Edges: j Edges: 30 Vertices: 9 Vertices: 6 Vertices: j Use Euler’s Formula to find the number of vertices in each polyhedron described below. 7. 6 square faces 8. 5 faces: 1 rectangle 9. 9 faces: 1 octagon and 4 triangles and 8 triangles Verify Euler’s Formula for each polyhedron.Then draw a net .
1 Articles 598-599 of Maxwell’s Treatise In his Treatise [55], Maxwell argued that an element of a circuit (Art. 598), or a particle (Art. 599) which moves with velocity v in electric and magnetic fields E and B μH experiences an electromotive intensity (Art. 598), i.e., a vector electromagnetic force given by eq. (B) of Art. 598 and eq. (10) of Art. 599,8
Freddie Mac Single-Family Seller/Servicer Guide Chapter 1101 As of 03/02/16 Page 1101-1 Chapter 1101: The Guide 1101.1: Introduction to the Guide (03/02/16) This section provides
Glossary of Useful Terms: pp. 647-659 ENGL 1101 1 GORDON STATE COLLEGE ENGL 1101: English Composition I - Syllabus and Course Policy - Fall 2019 Part I - Introduction and Course Outline ENGL 1101 - CRN 307 - 8:00 AM - Rm IC 205 - MW Office Hours: - By appointment
Proving Triangles Congruent Lesson 4-1: Example 4 Extra Skills, Word Problems, Proof Practice, Ch. 4 PowerPoint Special Needs Have students mark congruent sides and angles as shown to distinguish between AAS and ASA. Belo
Isosceles and Equilateral Triangles Lesson 3-4 1. Name the angle opposite . lC 2. Name the angle opposite . lA 3. Name the side opposite &A. 4. Name the side opposite &C. 5. Algebra Find the value of x. 105 New Vocabulary legs of an isosceles triangle base of an isosceles triangl
Mar 01, 2013 · Lesson 12-4 Angle Measures and Segment Lengths 687 Angle Measures and Segment Lengths A is a line that intersects a circle at two points. is a secant ray, and is a secant segment. Angles formed by secants, tangents and chords intercept arcs on circles.The measures of the intercepted arcs can help you find the measures of the angles.File Size: 688KB
Find the perimeter and area of each figure. 1. 2. 3. Find the perimeter and area of each rectangle with the given base and height. 4. b 1 cm, h 3 cm 5. b 2 cm, h 6 cm 6. b 3 cm, h 9 cm 8 cm 6 cm 8 m 4 m 7 in. What You’ll Learn To find the perimeters and areas of similar figures. . .And Why To find the expected yield of a garden .
hardware IP re-use and consistency accross product families and higher level programming language makes the development job far more convenient when dealing with the STM32 families. HIGH-PERFORMANCE HIGH DEGREE OF INTEGRATION AND RICH CONNECTIVITY STM32H7: highest performance STM32 MCUs with advanced features including DSP and FPU instructions based on Cortex -M7 with 1 to 2 Mbytes of .
