3-7 Slopes Of Parallel And Perpendicular Lines
3-7ALGEBRASlopes of Parallel andPerpendicular Lines3-71. PlanObjectives12To relate slope and parallellinesTo relate slope andperpendicular linesExamples123456Checking for Parallel LinesDetermining Whether Linesare ParallelWriting Equations of ParallelLinesChecking for PerpendicularLinesWriting Equations forPerpendicular LinesReal-World ConnectionWhat You’ll LearnCheck Skills You’ll Need To relate slope and parallelFind the slope of the line through each pair of points.linesperpendicular linespage 165 and Lesson 3-62. H(0, -5), D(2, 0) 253. E(1, 1), F(2, -4) 55. x y 206. 2x - 3y 6 329. y 32x 7 23Find the slope of each line. . . And Why4. y 2x - 5To write an equation thatmodels part of a leaded glasswindow, as in Example 67. x y2 18. y 7 011Slope andand ParallelParallel LinesLines1 1 SlopeßPartThe relationship between slope and parallel lines is summarized below and provedin Lesson 7-3.Math BackgroundSlope is a fixed ratio thatcharacterizes any nonverticalline. It is another example of apure geometric concept, a line,described by algebraic methods.The product of the slopes ofperpendicular lines being -1,although initially surprising, ismerely the Pythagorean Theoremin an analytic geometry setting.1. F(2, 5), B(-2, 3) 12 To relate slope andGO for HelpKey ConceptsSummarySlopes of Parallel LinesIf two nonvertical lines are parallel, their slopes are equal.If the slopes of two distinct nonvertical lines are equal, the lines are parallel.Any two vertical lines are parallel.Real-WorldConnectionThe ramp and rails areparallel because they havethe same slope.You can test whether nonvertical lines are parallel by comparing their slopes.1More Math Background: p. 124DEXAMPLEChecking for Parallel LinesyAre lines O1 and O2 parallel? Explain.Lesson Planning andResources45 2 (24)9slope of O1 1 2 (22) 3 3See p. 124E for a list of theresources that support this lesson.slope of O2 PowerPointQuick CheckCheck Skills You’ll NeedFor intervention, direct students to:Slope174(3, 3)2 23 2 (24)73 2 1 2O 1 2( 2, 4)Lines O1 and O2 are not parallel becausetheir slopes are not equal.Bell Ringer PracticeAlgebra 1 Review, page 165(1, 5)Find and compare the slopes of the lines.13x 2(1, 4)1 Line O3 contains A(-4, 2) and B(3, 1). Line O4 contains C(-4, 0) and D(8, -2).Are O3 and O4 parallel? Explain. No; the slope of 3 17 , and the slopeof 4 16 .Chapter 3 Parallel and Perpendicular LinesGraphing LinesLesson 3-6: Example 3Extra Skills, Word Problems, ProofPractice, Ch. 3174Special NeedsBelow LevelL1Some students may have difficulty seeing pairs ofperpendicular lines other than vertical and horizontallines. Have students use a protractor to confirm theperpendicular lines shown in Example 4.learning style: tactileL2Review the rules for multiplying and dividing signednumbers before students work with the slopes ofperpendicular lines.learning style: verbal
2. TeachSlope-intercept form allows you to compare slopes easily in order to decidewhether lines are parallel.2145AABBCy 5 3x 2 1EDCy 5 2 13x 2 1y 5 13x 1 1EDCBEDCBA3CBA2DDGuided InstructionMultiple Choice Which line is parallel to 4y 2 12x 5 20?EDCBADetermining Whether Lines are ParallelEXAMPLEEETest-Taking Tip4y 5 212x 1 20To compare slopes oftwo lines, write theirequations in slopeintercept form,y 5 mx 1 b.CareersMost buildings have walls thatare perpendicular to floors, toceilings, and to each other.Have students investigate howbuilders construct these.Write 4y - 12x 20 in slope-intercept form.4y - 12x 204y 12x 20Add 12x to each side.y 3x 5Divide each side by 4.The line 4y 2 12x 5 20 has slope 3. The line y 5 3x 2 1 is the only answer choicewith slope 3. The correct choice is A.Quick Check2 Are the lines parallel? Explain.a. y 2 12 x 5 and 2x 4y 9Yes; both slopes – 12.b. y 2 12 x 5 and 2x 4y 20No; the lines are the same.You can write an equation for a line parallel to a given line.3EXAMPLEMath TipAfter students read the KeyConcepts about parallel lines,ask: How can you state the twoconditionals as a biconditional?Two distinct nonvertical lines areparallel if and only if their slopesare equal.2Writing Equations of Parallel LinesEXAMPLEError PreventionRemind students that they mustcompare both the slopes and they-intercepts. Ask: If the slopesare equal and the y-interceptsare equal, are the lines parallel?Explain. No; there is onlyone line.Write an equation for the line parallel to y -4x 3 that contains (1, -2).Step 1 Identify the slope of the given line.y -4x 3cslopeStep 2 Use point-slope form to write an equation for the new line.y - y1 m(x - x1)y - (-2) -4(x - 1)y 2 -4(x - 1)Quick CheckPowerPointSubstitute -4 for m and (1, –2) for (x1, y1).Additional ExamplesSimplify.1 Line 1 contains P(0, 3) andQ(-2, 5). Line 2 contains R(0, -7)and S(3, -10). Are lines 1and 2 parallel? Explain. Yes;each has slope –1 and differenty-intercepts.3 Write an equation for the line parallel to y -x 4 that contains (-2, 5).y –x 32Part1 2 Slope and Perpendicular Lines2 Are the lines y -5x 4 andx -5y 4 parallel? Explain.No; one line has slope –5 andthe other line has slope –15.The relationship between perpendicular lines and their slopes is summarizedbelow. These statements will be proved in Lessons 6-6 and 6-7.Key ConceptsSummarySlopes of Perpendicular Lines3 Write an equation in pointslope form for the line parallel to6x - 3y 9 that contains(-5, -8). y 8 2(x 5)If two nonvertical lines are perpendicular, the product of their slopes is -1.If the slopes of two lines have a product of -1, the lines are perpendicular.Any horizontal line and vertical line are perpendicular.Lesson 3-7 Slopes of Parallel and Perpendicular LinesAdvanced Learners175English Language Learners ELLL4After students complete Example 5, have them work onthis problem: Line y ax b is perpendicular to liney - ax b. What are the possible values of a?learning style: verbalReview the meaning of positive slope and negativeslope. Then distinguish slope from the product of theslopes of two lines. The product of the slopes for twononvertical perpendicular lines is always 1.learning style: verbal175
Guided Instruction4EXAMPLEYou can test whether lines are perpendicular by first noting whether either line isvertical or horizontal. If not, check their slopes. If the product of the slopes is -1,the lines are perpendicular.Alternative MethodWhen lines are perpendicular,their slopes are negativereciprocals. Have students checkthat m1 and m2 are negativereciprocals.5EXAMPLESpecial Needs4EXAMPLEChecking for Perpendicular LinesAlgebra Lines O1 and O2 are neither verticalnor horizontal. Are they perpendicular? Explain.Step 1 Find the slope of each line.22 2 2244m1 slope of O1 23 2 0 23 33 2 (23)63m2 slope of O2 22 2 6 28 24L1Students who find the procedurecomplicated can work withpartners.6EXAMPLEStep 2 Find the product of the slopes.34m1 ? m2 3 ? 24 -1DiversityQuick CheckPowerPoint4 Are O3 and O4 perpendicular? Explain.No; the slope of 3 94 , and the slope of 4 –37 , and 94 –73 u 1.Additional Examples 4O(0, 2)2 2( 3, 2)46x(6, 3)(0, 5)4y/3(5, 5)224x 2 (3, 2) /4N(4,-6). Line 2 contains P(-2, 9)and Q(5, 7). Are lines 1 and 2perpendicular? Explain. No; theproduct of their slopes is 1.You can write an equation for a line perpendicular to a given line. If the given lineis horizontal, write an equation for a vertical line. If the given line is vertical, writean equation for a horizontal line.5 Write an equation for a lineperpendicular to 5x 2y 1 thatcontains (10, 0). y 25(x – 10)56 The equation for a lineEXAMPLEWriting Equations for Perpendicular LinesWrite an equation for the line perpendicular to y -3x - 5 that contains (-3, 7).containing a lead strip isy 12x - 9. Write an equationfor a line perpendicular to it thatcontains (1, 7). y –2x 9Step 1 Identify the slope of the given line.y -3x - 5cslopeVocabulary TipNumbers with product -1are opposite reciprocals.In Example 5, theopposite reciprocalof -3 is 13.Step 2 Find the slope of the line perpendicular to the given line.Let m be the slope of the perpendicular line.The product of the slopes of perpendicular lines is –1.-3m -11m 3Divide each side by –3.Step 3 Use point-slope form to write an equation for the new line.y - y1 m(x - x1)1y - 7 3 [x - (-3)]1y - 7 3 (x 3)Closure1763( 4, 1) 4 2 O4 Line 1 contains M(0, 8) andExplain algebraically why twolines cannot be both parallel andperpendicular. The product ofthe slopes m of two parallel linesis m2. The product of the slopesof two perpendicular lines is –1.A slope m such that m2 –1 isnot a real number.( 2, 3) 1Lines O1 and O2 are perpendicular because the product of their slopes is -1.Ask students to describe unusualwindow designs they have seenelsewhere.Resources Daily Notetaking Guide 3-7 L3 Daily Notetaking Guide 3-7—L1Adapted Instructiony 2Quick Check176Substitute 13 for m and (–3, 7) for (x1, y1).Simplify.5 Write an equation for the line perpendicular to 5y - x 10 that contains (15, -4).y 4 5(x 15)Chapter 3 Parallel and Perpendicular Lines
6BAReal-WorldEXAMPLE3. PracticeConnectionThe window at the left includes some perpendicular lead strips. The linethat contains BC has equation y -x 10. AB is perpendicular to BC.* )Write an equation for AB , the line that contains AB and point (-1, 5).CDAssignment Guide1 A B 1-15, 29-32, 34, 35, 37* )The line that contains BC has slope -1. Let m be the slope of AB .-1m -1* )2 A BThe product of the slopes is –1.C Challengem 1AB has slope 1 and can be written in the form y 1x b, or y x b.Test PrepMixed Review* )y x bAB has slope 1.5 -1 bSubstitute 5 for y and –1 for x.6 bAdd 1 to each side.* )To check students’ understandingof key skills and concepts, go overExercises 6, 18, 34, 36, 40.6 If the equation for a line containing a lead strip on a different window isy 2 23x 15, write an equation for the line perpendicular to it that contains (2, 8).y 8 EXERCISES32 (x 2) or y 32xFor more exercises, see Extra Skill, Word Problem, and Proof Practice.Practice by ExampleExample 1GO forHelp(page 174)2. No; the slope of 1 31and the slope of 2 12 .In Exercises 1–5, are lines 1 and 2 parallel? Explain, using slope. 2–3. See left.Yes; both slopes 12 .1.2.yy(0, 3) 2(1,3)2( 5, 1) 1 2(4, 2)( 6, 1)(4, 1) 2xxO 2 6 44O 624 1(0, 2)( 4, 2)3.4.y32,3. No; the slope of 1 and the slope of 2 2.(5, 3)3(0, 2) 4 2O 1 2( 4, 4)2x4Visual LearnersExercises 1–4 Lines that do not 5Practice and Problem SolvingA49-5253-61Homework Quick CheckThe equation for AB is y x 6.Quick Check16-28, 33, 36, 38-4445-48Yes; both slopes 4.y( 2, 5)4 2 1 2( 3, 1)(1, 2)GPS Guided Problem SolvingO24L3L4EnrichmentL2ReteachingL1Adapted PracticeName 2 2(2, 3)Exercises 8–11 Remind studentsto rewrite equations in slopeintercept form.Practice 4 2intersect on the portion of thecoordinate plane shown or thatappear parallel may not actuallybe parallel. The only way to becertain that lines are parallel isto compare their slopes.xClassL3DatePractice 3-7Constructing Parallel and Perpendicular LinesConstruct a line perpendicular to line l through point Q.1.(0, 2)2.Q3.QQ (page 175)6. y 2x 5y 2x9. y - 7x 6y 7x 87. y 34x - 10y 34x 28. y -x 610. 3x 4y 1211. 2x 5y -16x 2y 6Construct a line perpendicular to line l at point T.4.T 5.T 6. TConstruct a line parallel to line l and through point K.7.8.K9.KK x y 20 Pearson Education, Inc. All rights reserved.Example 25. Line 1 contains A(-3, 6) and B(2, 6), and line 2 contains C(0, 0) and D(7, 0).Yes; both slopes 0.x 2 Algebra Are the lines parallel? Explain. 6–11. See margin.10y -4x - 20For Exercises 10–15, use the segments at the right.10. Construct a quadrilateral with one pair of parallel sides of lengthsa and b.abc11. Construct a quadrilateral with one pair of parallel sides of lengthsb and c.12. Construct a square with side lengths of b.13. Construct a right triangle with leg lengths of a and c.14. Construct a right triangle with leg lengths of b and c.15. Construct an isosceles right triangle with leg lengths of a.Lesson 3-7 Slopes of Parallel and Perpendicular Lines6. Yes; the lines both havea slope of 2 butdifferent y-intercepts.7. Yes; the lines both havea slope of 34 butdifferent y-intercepts.8. Yes; the lines both havea slope of 1 butdifferent y-intercepts.1779. No; one slope 7 andthe other slope 7.10. No; one slope 34 and11. Yes; the lines both havea slope of 25 butdifferent y-intercepts.the other slope 3.177
Exercise 28 Students mustExercise 38 Discuss as a class howthe truth value of this theoremchanges if the phrase in a planeis removed. Have students explainwhy the resulting statement isfalse.Exercise 39 In Chapter 5 studentswill learn that the shortestdistance from a point to a lineis along the perpendicular pathto that line. Discuss informallywhy Joe would choose theperpendicular route to the ball.* )Write an equation for the line parallel to AB that contains point C.* )* )SeeSee(page 175)12. AB : y -2x 1, C(0, 3)13. AB : y 13x, C(6, 0) margin.margin.* )* )14. AB : -x1 2y 4, C(-2, 4)15. AB : 3x 2y 12, C(6, -2)y 4 2 (x 2)y 2 32 (x 6)2Example 4 x Algebra Are lines O1 and O2 perpendicular? Explain using slope.Example 3correctly transform equations instandard form to slope-interceptform.(page 176)16.y16. Yes; the slope of 1 21 , and the slopeof 2 2; 12 2 1.1Exercises 45, 46 These exercisesanticipate the study of quadrilaterals in Chapter 6. If necessary,review the distance and midpointformulas.19. Yes; the slope of 1 1, and the slopeof 2 1; 1 1 1.O 1213 (x13. y 0 y 13 (xExample 5(page 176) 4 6)Example 629. slope of AB slope ofCD 23 ; AB zz CDslope of BC slope ofAD 3; BC zz AD30. slope of AB slope ofCD 34 ; AB zz CDslope of BC slope ofAD 1; BC zz AD31. slope of AB 12 ; slopeof CD 14 ; AB zz CDslope of BC 1; slopeof AD 12 ; BC zz AD32. slope of AB slope ofCD 0; AB zz CDslope of BC 3 andslope of AD 32 ;BC zz AD(page 177) "(7 21 (11 21) 2 "104BD: d "(13 2 3) 2 1 (7 2 5) 2 "104AC O BD178 418. 1 4 2O( 2, 4)2 2O5(1, 1) 4x 2(3, 3) 22x4(0, 4)19.(6, 3)(4, 3)2( 5, 3)y( 4, 4) 24y(3, 4) 1(2, 0)4Ox( 3, 2) 4)*Write an equation for the line perpendicular to MN that contains point P.y 2(x 4)* )* )y 6 32 (x 6)20. MN : y 23x, P(6, 6)21. MN : y 12x - 5, P(4, 0))22. MN : y 2x -8, P(4, 4)y 4 12 (x 4)24. Highway Construction Highwayplanners want to construct a roadperpendicular to Route 3 at point O.An equation for the Route 3 line isy 23x. Find an equation for theline for the new road. y 32 x*)23. MN : 4y 5x 20, P(0, 0)y 45 xNy2Route 3W 2O2x E 2SApply Your Skills x 2 Algebra Are the lines perpendicular? Explain.No; 72 ( 74 ) u 1.Yes; one is vertical and25. y -x - 726. y 3 the other is27. 2x - 7y -42y - x 20x -2 horizontal.4y -7x - 2Yes; 1 ( 1) 1.28. Multiple Choice Which line is perpendicular to 3y 1 2x 5 12? A6x - 4y 24y 3x -22x 3y 6y -2x 633. Answers may vary.Sample: y 54 x 5,Use slopes to find whether the opposite sides of quadrilateral ABCD are parallel.y 54 x 529–32. See margin.29. A(0, 2), B(3, 4), C(2, 7), D(-1, 5)30. A(-3, 1), B(1, -2), C(0, -3), D(-4, 0)BGOnlineHomework HelpVisit: PHSchool.comWeb Code: aue-030745. AC: d9) 2 1 ( 4, 2)x(1, 5)* 6) ory(4, 4) 412. y 3 2(x 0) ory 3 2x17.(4, 1)4(0, 2)17. Yes; the slope of 1 23 ,and the slope of 2 23 ; 32 23 1.18. No; the slope of 1 1,and the slope of 2 45 ; 1 45 u 1. 217831. A(1, 1), B(5, 3), C(7, 1), D(3, 0)33. Open-Ended Write equations for two perpendicular lines that have the samey-intercept and do not pass through the origin. See left.34. Writing Can the y-intercepts of two parallel lines be the same? Explain.No; two parallel lines with the same y-intercept are actually only one line.Chapter 3 Parallel and Perpendicular Lines46. slope of AC 5;slope of BD 15 ;since 5 ? 15 1,AC ' BD;32. A(1, 0), B(4, 0), C(3, -3), D(-1, -3)midpoint AC (8, 6);midpoint BD (8, 6);since the midpoints arethe same, the diagonalsbisect each other.
35. RS and VU arehorizontal with slope 0; RS zz VU; slopeof RW slope of UT 1; RW zz UT ; slopeof WV slope of ST 1; WV zz ST4. Assess & Reteach35. Use slope to show that the opposite sides ofhexagon RSTUVW at the right are parallel.36. Use slope to determine whether a triangleGPS with vertices G(3, 2), H(8, 5), and K(0, 10) is aright triangle. Explain. No; no pairs of slopeshave a product of 1.Developing Proof Use slope to explain why eachtheorem is true for three lines in the coordinate plane.PowerPointLesson QuizOExercise 3537. Theorem 3-9: If two lines are parallel to the same line, then they are parallel toeach other. The lines will have the same slope.Goal38. Theorem 3-10: In a plane, if two lines are perpendicular to the same line, thenthey are parallel to each other. When lines are #, the product of theirslopes is 1. So, two lines # to the same line must have the same slope.39. Soccer The coordinate system at the rightO10 20 30 40 xis designed for a soccer field. Each unit 10represents one yard. Joe is at point P(35, -20).The path of the ball from a corner kick isJoe 20represented by the equation y 2 43x. Tohave the best chance for a shot on goal, Joe 30wants to run toward the ball so that his pathmeets the path of the ball at a right angle.a. Find an equation for the line on which Joeyshould run. y 20 34(x 35)b. Critical Thinking Why is point-slope form the best choice for the equation?because you are given a point and can quickly find the slope* ) * )Determine whether AB and CD are parallel, perpendicular, or neither.Real-WorldConnectionFor a corner kick, the ball isplaced within a quarter circleof radius 1 yd.40. A Q -1, 12 R , B(-1, 2), C(3, 7), D(3, -1) n 41. A(-2, 3), B(-2, 5), C(1, 4), D(2, 4) #42. A(2, 4), B(5, 4), C(3, 2), D(0, 8)neitherPHSchool.comFor: Graphing calculatorproceduresWeb Code: aue-2122CChallenge43. A(-3, 2), B(5, 1), C(2, 7), D(1, -1) #* )44. Graphing Calculator Use your graphing calculator to find the slope of ABin Exercise 43. Enter the x-coordinates of A and B into the L1 list of yourlist editor. Enter the y-coordinates into the L2 list. In yourCALCABD47.a – b. Answers may vary.Sample:C5x10Exercises 45 and 46Lesson 3-7 Slopes of Parallel and Perpendicular Linesy6RQ42OP24S (6, 1)x6(0, 4)xO(4, –3) 2(0, –5)Yes; the lines havethe same slope and differenty-intercepts.2. Are the lines x 4y 8 and 2x 6y 16 parallel? Explain.No; their slopes are not equal.3. Write an equation in pointslope form for the line parallelto -18x 2y 7 thatcontains (3, 1).y – 1 9(x – 3)5. Write an equation inpoint-slope form for theline perpendicular toy -16x - 2 that contains(-5, -8). y 8 6(x 5Alternative Assessment48. A triangle has vertices L(-5, 6), M(-2, -3), and N(4, 5). Write an equation forthe line perpendicular to LM that contains point N. y 5 13(x 4)lesson quiz, PHSchool.com, Web Code: aua-0307(–2 , 3)4. Are the lines y 32x 5and 3x 2y 10 perpendicular? Explain. Yes; theproduct of their slopesis –1.menu select LinReg (ax b).to find the slope a. Repeat to find the* )* ) * )slope of CD . Are AB and CD parallel, perpendicular, or neither? #45. Show that the diagonals of theyfigure at the right are congruent.10See margin, p. 178.46. Show that the diagonals of the figureat the right are perpendicularbisectors of each other. See margin,p.178.547. a. Graph the points P(2, 2), Q(7, 4),and R(3, 5). a–c. See margin.b. Find the coordinates of a point Sthat, along with points P, Q, andR, will form the vertices of aOquadrilateral whose opposite sidesare parallel. Graph the quadrilateral.c. Repeat part (b), finding a differentpoint S and graphing the new quadrilateral.1. Are lines 1 and 2 parallel?Explain. 1y179c. The other possiblelocations for S are( 2, 3) and (8, 7).Have students draw a pair ofparallel lines and a pair of perpendicular lines on a coordinateplane and then use slopes toprove that the lines are paralleland perpendicular.Test PrepResourcesFor additional practice with avariety of test item formats: Standardized Test Prep, p. 193 Test-Taking Strategies, p. 188 Test-Taking Strategies withTransparencies179
Test PrepStandardizedTest PrepUse this Checkpoint Quiz to checkstudents’ understanding of theskills and concepts of Lessons 3-5through 3-7.49. What is the slope of a line parallel to the line 6x - 4y 12? B3A. 23B. 2C. 4D. 24233Multiple Choice50. The slope of a line is 6. What is the slope of a line perpendicular to it? JF. 6G. -6H. 1J. 2166Resources51. Line f contains the points (5, -4) and (4, -6). What is the slope of a lineperpendicular to it? CA. 2B. 1C. 21D. -222Grab & Go Checkpoint Quiz 252. Line c contains the points (2, -2) and (-4, 1). a–b. See margin.a. What is the slope of a line perpendicular to line c?b. What is the y-intercept of the line perpendicular to line c thatcontains (1, 2)?Short ResponseMixed ReviewGO forHelpLesson 3-6 x 2 Algebra Write an equation for the line containing the given points.53. A(0, 3), B(6, 0)54. C(-4, 2), D(-1, 7)55. E(3, -2), F(-5, -8)y 3 21(x 0) or y 3 12 xy 2 35(x 4)y 2 34(x 3)Lesson 2-4Name the property that justifies each statement.Mult. Prop. of 56. &4 &4 Refl. Prop. of O57. If m&B 8, then 2m&B 16.58. -3x 6 3(-x 2) Dist. Prop.52. [2] a. slope of line c:1 2 (22)24 2 2 53126 5 22 ; slopeLesson 2-3Use the Law of Syllogism to draw a conclusion.of60. If you are in geometry class, thenyou are in math class. If you are inmath class, then you are at school.If you are in geometry class,then you are at school.line # to c: 2b. 0[1] at least one correctslope4.x 4 O1.2 8 105.647. slope of RS 25;4slope of TV 53; neither48. slope of RS 1;4slope of TV 1; #49. slope of RS 54;4slope of TV 45; #y282xy42O180n (n 30) 134 (n 12) (n 16) 161 180x2 4 6 8 102.58 x w 3.50 89 (2w 25) (x 6) a m 64 100 quad; a 105;octagon; n 125hexagon; x 122; w 90m 1162x Algebra Graph each line using intercepts. 4–6. See margin.n 4 O6.Lessons 3-5 through 3-7Use the number of sides to name the polygon. Then find the value of each variable.y261. If you travel to Switzerland, then youtravel to Europe. If you travel toEurope, then you have a passport.If you travel to Switzerland,then you have a passport.CheckpointCheckpoint QuizQuiz 22Checkpoint Quiz 259. If RS MN, then MN RS.Symm. Prop. of O126 4. 4x y -8* )125 5. -2x 3y 12* )6. 3x 5y 30* )* )Find the slopes of RS and TV. Then determine whether RS and TV are parallel,perpendicular, or neither. Explain.7. R(-2, 6), S(3, 4), T(3, 5), V(0, 0)8. R(6, -1), S(7, 0), T(3, -4), V(0, -1)9. R(9, 1), S(5, 6), T(3, 8), V(-2, 4)10. R(5, -7), S(-4, -9), T(6, 2), V(-3, 0)44slope of RS 29; slope of TV 29; nChapter 3 Parallel and Perpendicular Lines
perpendicular lines. . .And Why To write an equation that models part of a leaded glass window, as in Example 6 3-7 11 Slope and Parallel Lines Key Concepts Summary Slopes of Parallel Lines If two nonvertical lines are parallel, their slopes are equal. If the slopes of two distinct nonvertical lines are equal, the lines are parallel.
Visit our Photo Gallery See Project Case Studies Design Resources for your project. VEGETATED SLOPES AGGREGATE SLOPES CONCRETE SLOPES Slope Protection Infill Options . Visit our Video Gallery Watch Cross-Section Animation Simple Installation Animations GEOWEB Slopes with Anchors Video GEOWEB Slopes with Tendons & Tendon Clips Video
Slopes of Parallel and Perpendicular Lines Example 4A: Writing Equations of Parallel and Perpendicular Lines Write an equation in slope-intercept form for the line that passes through (4, 10) and is parallel to the line described by y 3x 8. Step 1 Find the slope of the line. y 3x 8
Step 1 Find an equation of a line perpendicular to the two parallel lines. The slopes of the two parallel lines are both 1 — . Use the Slopes of Perpendicular Lines 2 Theorem. 1 — 2 m 1 The product of the slopes of lines is 1. m 2 Multiply each side by 2. Any line with a slope of 2 is perpendicular to the two .
This engineer manual (EM) provides guidance for analyzing the static stability of slopes of earth and rock-fill dams, slopes of other types of embankments, excavated slopes, and natural slopes in soil and soft rock. Methods for analysis
Any two vertical lines are parallel. Postulate 18 Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. The slopes of the two lines that are perpendicular are negative reciprocals of each other. Horizontal lines are perpendicular to vertical lines
IXL Practice Worksheets Creating S18 – (Algebra 1) (at least to 85) Parallel Slopes Level 2 Create a graph using www.desmos.com of 10 parallel lines. Take a screenshot of the graph, upload it to google drive and share it with me. 4. Take the Schoology Quiz (Concept 8: Level 2) Score of
3 Parallel and Perpendicular Lines Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. 3.1 Pairs of Lines and Angles 3.2 Parallel Lines and Transversals 3.3 Proofs with Parallel Lines 3.4 Proofs with Perpendicular Lines 3.5 Slopes of Lines 3.6 Equations of Parallel and .
Introduction to Description Logic Szymon Klarman (part of the content based on the tutorial by Stefan Schlobach) szymon.klarman@gmail.com VU University Amsterdam, 2009-2012. AR@AI Introduction to Description Logic Plan for today Tableau algorithm for ALCwith empty TBoxes Soundness, completeness, termination Reasoning w.r.t. non-empty TBoxes Szymon Klarman 1 / 1. AR@AI Introduction to .
