7-1 Ratios And Proportions - Portal.mywccc

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7-1ALGEBRA7-1Ratios and Proportions1. PlanObjectives1To write ratios and solveproportionsExamples1234Real-World ConnectionProperties of ProportionsSolving for a VariableReal-World ConnectionWhat You’ll LearnCheck Skills You’ll Need To write ratios and solveSimplify each ratio.proportions. . . And WhyTo find dimensions from ascale drawing, as in Example 4More Math Background: p. 364CSkills Handbook p. 756 and Lesson 5-11. 24 128 22. 1233. 68 344. 10105. 20 30 236. 8 to 2 47. 2 to 8 148. 12 9 4319. Draw a triangle. Then draw its three midsegments to form a smallertriangle. How do the lengths of the sides of the smaller triangle compare tothe lengths of the sides of the larger triangle?Each side of the smaller k is 12 the length of a side of the larger k.New Vocabulary ratio proportion extended proportion Cross-Product Property scale drawing scaleMath BackgroundThe properties of proportionsare variations of applying theMultiplication and AdditionProperties of Equality. Thus thesame units must be within eachratio or in comparable positionsin a proportion.GO for Help1Using Ratios and ProportionsA ratio is a comparison of two quantities. You can write the ratio of a to b or a bas the quotient ba when b 2 0. Unless otherwise stated, the terms and expressionsappearing in ratios in this book are assumed to be nonzero.Vocabulary TipYou can read a b asthe ratio a to b.1Lesson Planning andResourcesEXAMPLEReal-WorldConnectionPhotography A photo that is 8 in. wide and 531 in. high is enlarged to a poster thatis 2 ft wide and 1 13 ft high. What is the ratio of the width of the photo to the widthof the poster?See p. 364E for a list of theresources that support this lesson.PowerPointBell Ringer PracticeCheck Skills You’ll NeedFor intervention, direct students to:Simplifying ExpressionsSkills Handbook, p. 756Midsegments of TrianglesLesson 5-1: Example 1Extra Skills, Word Problems, ProofPractice, Ch. 5width of photo8 in.8 in.81width of poster 2 ft 24 in. 24 3The ratio of the width of the photo to the width of the poster is 1 3 or 13.Quick Check3661 What is the ratio of the height of the photo to the height of the poster? 1 : 3Chapter 7 SimilaritySpecial NeedsBelow LevelL1Have each student make a scale drawing of theirbedroom as shown in Example 4. Students need tomeasure their room and use graph paper to maketheir drawings. Each drawing must include a scale.366learning style: tactileL2Have students make a list of equivalent fractions andtest them for equivalence by applying the propertiesof proportions.learning style: verbal

A proportion is a statement that two ratios are equal. You can write a proportionin these forms:Vocabulary TipabYou can read both 5and a b c das a is to b as c is to d.cd2. Teachacb d and a b c dWhen three or more ratios are equal, you can write an extended proportion. Forexample, you could write the following:61424 16 4Guided Instruction1Two equations are equivalent when either can be deduced from the other using theProperties of Equality. Several equations are equivalent to a proportion. Some ofthem are important enough to be called Properties of Proportions.PropertyProperties of Proportionsacb d is equivalent toEXAMPLE(1) ad bc(2) ba dcb(3) ac dc1db(4) a 1b dy6x 5meansTTa b c dextremesad bcd.x y65x1y7y 7x1y11y 62 Complete: If a 12 , then4bb.Cross-Product Propertyna. 52 35141xb. x 1 32 24 Two cities are 31 in. apart on a2map with the scale 1 in. 50 mi.Find the actual distance. 175 miy13y8 44(y 3) 8y7x 604y 12 8yx 60712 4yResources Daily Notetaking Guide 7-1 L3 Daily Notetaking Guide 7-1—L1Adapted Instructiony 33 Solve each proportion.a. 5z 203 0.75Closure6b. n 1816 n 3Lesson 7-1 Ratios and ProportionsAdvanced LearnersAsk: Ifab cd, ?. a43 Solve each proportion.Solving for a Variablea. x5 1277x 5(12)1 A scale model of a car is 4 in.long. The actual car is 15 ft long.What is the ratio of the length ofthe model to the length of the car?1 : 45b12Algebra Solve each proportion.Quick CheckPoint out that the scale comparesinches to feet, so each ratiohas inches in the numeratorand feet in the denominator.PowerPointYou solve a proportion by finding the value of the variable.EXAMPLEEXAMPLEcab d2 Critical Thinking Write two proportions that are equivalent to m n .4114m 1 411Answers may vary. Sample: m 11 n1n,4113Teaching TipAdditional ExamplesAlgebra If xy 56, complete each statement.ya. 6x 7b. x 7c. x5 777Quick Check4Properties of Proportions6x 5yEXAMPLEReplace the variables withnumbers to verify that theproportions are equivalent.Multiplying both sides of ba dc by bd results in the firstproperty, called the Cross-Product Property. You may statethis property as “The product of the extremes is equal to theproduct of the means.”2Math TipStudents should understand thatunits of measurement must bethe same.2Key ConceptsEXAMPLE367English Language Learners ELLL4222does a2 equal dc 2? Does ba equal a2?bbYes; yes, only if a b or a 0learning style: verbalA baseball batting average is theratio of hits to at-bats, expressedas a decimal. If a player with 540at-bats has a batting average of0.350, how many hits did theplayer make? 189 hitsPoint out that a proportion involves two ratios thatare equal. Sometimes the term proportion is usedincorrectly for a ratio, such as “the proportion ofstudents who have cell phones is 3 out of 5.”learning style: verbal367

3. PracticeIn a scale drawing, the scale compares each length in the drawing to the actuallength. The lengths used in a scale can be in different units. A scale might bewritten as 1 in. to 100 mi, 1 in. 12 ft, or 1 mm i 1 m. You can use proportions tofind the actual dimensions represented in a scale drawing.Assignment Guide1 A B 1-55C Challenge456-61Test PrepMixed Review62-6667-775ABBHomework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 16, 22, 26, 34, ctionEDDEETest-Taking TipIf you use a ruler andyour answer does notmatch any answerchoice, first check thatyou measuredcorrectly.Multiple Choice Use a ruler to measure the lengthO of the bedroom in the scale drawing. What is thelength of the actual bedroom?10 ft18 27 ft24 52 ftᐉAn inch ruler shows that / 5 78 in.71 816/drawing length (in.)actual length (ft)O 16 Q 78 RError Prevention!14 ftwScale: 1 in. 16 ftCross-Product PropertyO 14Exercise 2 Ask: Why is writing theThe actual bedroom is 14 ft long. The correct answer is B.6185ratio asincorrect? 185 ft firstmust be converted to inches, or6 in. to feet.Quick Check4 Find the width of the actual bedroom. 10 ftAuditory LearnersExercises 3–11 Have studentsEXERCISESexplain their reasoning aloudto the class, citing the Propertyof Proportions.Practice and Problem SolvingAPractice by ExampleExample 1GO forHelpGPS Guided Problem Solving(page 366)L4L2ReteachingL1Adapted PracticePracticeNameClassL3DatePractice 7-1Areas of Parallelograms and TrianglesFind the area of each triangle, given the base b and the height h.1. b 4, h 42. b 8, h 23. b 20, h 64. b 40, h 125. b 3.1, h 1.76. b 4.8, h 0.87. b 3 41 , h 128. b 8, h 2 141. The base of the pyramid at theright is a square whose sidesmeasure 0.675 m. The intent wasfor the sides to measure 675 m.What is the ratio of the length ofa base side in the small pyramidto the length of a base side in theintended pyramid? 1 : 10002. Models The Leaning Tower ofPisa in Italy is about 185 ft tall.A model of the Leaning Toweris 6 in. tall. What is the ratio ofthe height of the model to theheight of the real tower?1 : 370L3EnrichmentFor more exercises, see Extra Skill, Word Problem, and Proof Practice.“We had a little problemwith the decimal point.”9. b 100, h 30Find the value of h in each parallelogram.10.11.h812.h7Example 2 x 2 Algebra If ab 34 , complete each statement.(page 367)43. 4a 7 3b4. ba 77 3b36. 43 77. b4 77 a7 aba 7 37 79. a 110. a 1b 7 4b7 72214h5410114.313. What is the area of ABCD with vertices A(-4, -6), B(6, -6),C(-1, 5), and D(9, 5)?14. What is the area of DEF with vertices D(-1, -5), E(4, -5),and F(4, 7)?Find the area of the shaded region.15.16.17.61151144513P ( 2, 3)414Find the area of each parallelogram.18.yD ( 3, 4)C (3, 4)y19. 2 2A ( 6, 4)36824 x 6 2 2K (3, 0)26xB (0, 4)T ( 1, 5)Q (6, 3)22H ( 5, 0) 6y20.42J (7, 5) 4 2 2S ( 4, 4)246xR (4, 4) Pearson Education, Inc. All rights reserved.6148368Chapter 7 Similarityb5. a3 77 48. 3b 7 4a11.a133b 1 4 77 4

Exercises 12–20 Remindstudents to check their answersby substituting the solution forx in the original proportion. Ask:How can you tell whether youranswer is correct? The proportionwith the substituted value is true.Example 3 x 2 Algebra Solve each proportion.(page 367)Example 4(page 368)12. x2 84 413. 95 x3 123x 414. 13 128 6.87515. x5 1116. x4 59 7.217. 56 x6 7.2310 1 4 7.518. x 13 471519. x 17 5 14620. 35 x 13 721. Geography The scale for this map ofLouisiana is 1 in. 40 mi. On themap, the distance from Lake Charles1125 mi to Baton Rouge is about 3 8 in. Abouthow far apart are the two cities?Exercise 21 Provide students withcopies of a map of your ownstate. Have them write and solveproblems similar to the one inExercise 21 and then exchangeand solve one another’s problems.Exercise 29 Ask: In what ordershould the ratio list the dinnerplate sizes? dollhouse to full-sizeExercise 50 The baskets on eachMap Reading Measure the map distance. Then find the actual distance.BApply Your Skills22. Morgan City to Rayne 23. Vinton to New Roads 24. Kaplan to Plaquemineabout 135 miabout 67.5 miabout 75 mi25. Design You want to make a scale drawing of your bedroom to help youarrange your furniture. You decide on a scale of 3 in. 2 ft. Your bedroom is a12 ft-by-15 ft rectangle. What should be its dimensions in your scale drawing?18 in. by 22.5 in.For each rectangle, find the ratio of the longer side to the shorter side.26.27.28.30 m18 in.50 in.13 : 665 m2 ft5:44:329. Multiple Choice The diameter of a dinner plate is 1 ft. In a dollhouse set, thediameter of a dinner plate is 114 in. What is the ratio of the diameter of thedollhouse plate to the diameter of the full-size plate? A45Complete each statement.y7 730. If x7 3 , then xy 7. 37 30t32. If 30 18r , then r 7. 18tGOnlineHomework HelpVisit: PHSchool.comWeb Code: aue-070154Connection to AlgebraExercises 56–58 You can dothese as class exercises by havingstudents provide steps andreasons in prooflike fashion.For each exercise, a possible firststep is using the Cross-ProductProperty.Exercises 59–61 If students need40 in.548end of a basketball court are 10 ftabove the floor. Have studentsexplain how they would representbaskets on their scale drawings.help getting started, suggestthat they break each extendedproportion into two proportions.4857 931. If 4m 9n, then mn 7. 47 ba5b1233. If a 15 2 , then 5 7. 234. Writing Use a map in your classroom or a map from a textbook. Explain howto use a ruler and the scale of the map to approximate an actual distance. Givean example. (If you do not have access to another map, use the map above.)Check students’ work.x 2 Algebra Solve each proportion.y35. 10 1525 69 12 3236. 24nb37. 1114 21 16.55 10 638. x 23x8 4 439. n 14nb 1240. 2b 52 1 12193241. x 23 x124, –342. 3 2 4x 2 11 3x1 1 5x512 , –6lesson quiz, PHSchool.com, Web Code: aua-0701Lesson 7-1 Ratios and Proportions369369

4. Assess & Reteach43. Models The sandwich shop at the left is 40 ft tall. The shop is an enlargementof an actual milk bottle. The scale used in construction is 5 ft 2 cm. Find theheight of the actual milk bottle. 16 cmPowerPoint44. Geography Students at the University of Minnesota in Minneapolis built aGPS model globe 42 ft in diameter using a scale of 1 1,000,000. About how tall isMount Everest on the model? (Mount Everest is about 29,000 ft tall.) 0.348 in.Lesson Quiz1. A scale model of a boat is9 in. long. The boat’s actuallength is 60 ft. Find the ratio ofthe length of the scale modelto the length of the boat.1 : 802. Solve the proportion10158 x . 123. A map uses the scale1 cm 20 mi. A county is 90mi wide. How wide is thecounty on the map?412 cmComplete each extended proportion.Real-WorldConnectionThis sandwich shop is onMuseum Wharf in Boston,Massachusetts.50. A basketball court is 92 ft by 50 ft.C? 1118c 1 2d55. a 1b 2b 5 jdj56. Property (2): If ba dc , then ba dc.b.57. Property (3): If ba dc , then ac dc1db58. Property (4): If ba dc , then a 1b d .Show the class a wall map of thecontinental United States, makingsure that you cover the scale. Askstudents to measure the widthof the United States on the mapfrom San Francisco, California, toWashington, D.C. Point out thatthe actual width of the UnitedStates is about 3000 mi. Havestudents use this informationto estimate the scale on themap. When they finish theirwork, uncover the scale so thatstudents can compare it withtheir answers.Solve each extended proportion for x and y with x S 0 and y S 0.y60. x5 y9 25x 3; y 15104x59. x6 x 118 yx 5; y 244 761. x1 x 19yx 3; y 21Test PrepMultiple ChoiceSolve each proportion.7 C62. 21 3x4 1 G63. x 21xx 2 D64. x 163x13 H65. 38 9Short Response23.266a. 2.7516 x (ORequivalent proportion)b. 135 km370A. 3F. -3A. 4F. 3 38B. 7C. 9D. 14G. 2 131H. 3J. 3B. 6C. 8D. 12G. 3H. 38J. 1366. A map of Long Island has the scale 2.75 cm 16 km. On the map, TargetRock is 23.2 cm from Lake Montauk.a. Write a proportion that you can solve to determine the actual distancefrom Target Rock to Lake Montauk. a–b. See left.b. Find the actual distance. Round your answer to the nearest kilometer.Chapter 7 Similarity49.48.1 c 5 j c or a54. ba 1bj ddChallenge x 2 Algebra Justify the indicated property of proportions. 56–58. See margin p. 421.Alternative Assessment48 – 51. Answers may vary.Samples are given.51. A football field is 160 ft by 120 yd.52. Error Analysis One rectangle has length 3 in. and width 4 ft. Another rectanglehas length 3 ft and width 4 yd. Elaine claims that the two rectangles are similarbecause their corresponding angles are congruent and their correspondingsides are in proportion. Explain why Elaine’s reasoning is incorrect.Elaine did not convert units, and thought the ratios equaled 1.caIf b 5 d , complete each statement.1 b 5 j b or a53. ca 1cj ddy4. x ? 1175. 7y ? 11xx 1 yy35j47. 14j 5 12 5 20 8; 21Games Choose a scale and make a scale drawing of the playing region.48–51. See margin.48. A pool table is 5 ft by 10 ft.49. A bowling lane is 3.5 ft by 60 ft.7xIf y 11, complete each of thefollowing.6.j 5 15 5 j 9; 1246. 1520258 5 6 5 12 9; 1845. 12jj50.51.Scale 1 cm 10 ftScale 1 in. 5 ft370Scale 1 in. 46 ftScale 1 cm 32 ft

Test PrepMixed ReviewLesson 6-7GO forHelpyPO(a, 0)For additional practice with avariety of test item formats: Standardized Test Prep, p. 411 Test-Taking Strategies, p. 406 Test-Taking Strategies withTransparencies68. parallelogram (a c, b)y(0, b)P67. square (a, a)Lesson 6-1ResourcesGive the coordinates for point P without using any new variables.(-c, 0) Ox(a, 0)x69.Graph each quadrilateral ABCD. Classify ABCD in as many ways as possible.69–72. See margin.69. A(-1, -2), B(3, -2), C(1, 4), D(-3, 4)yD270. A(2, -1), B(6, 2), C(8, 2), D(10, -1)71. A(-7, 1), B(-5, 3), C(0, -2), D(-2, -4) 2 OI. #PQR is isosceles. I and IIIII. #PQR is an obtuse triangle.III. #PQR is scalene.74.BparallelogramIn each exercise, identify two statements that contradict each other.73.x2A72. A(1, 1), B(-4, 4), C(1, 7), D(6, 4)Lesson 5-4CI. &1 &2 II and IIIII. &1 and &2 are complementary.III. m&1 m&2 18070.yB2O 2Write (a) the inverse and (b) the contrapositive of each statement. 75–77.See margin.75. If an angle is acute, then it has measure between 0 and 90.xADtrapezoid71.76. If two lines are parallel, then they are coplanar.C77. If two angles are complementary, then both angles are acute.yB2A 6 4 2 O 2 CxDA P int in Time15001600170018001900CD4 4 OJupiterused proportions to estimate the speed ofSunlight. He carefully measured the movementsAof Jupiter’s moons. With Earth at point B,EarthMoona moon emerged from behind Jupiternot drawn to scale16.6 min later than when Earth was at point A.He reasoned that it must have taken 16.6 minfor the light to travel from point A to point B. Using proportions, Römerestimated the speed of light to be 150,000 mi/s. This estimate is about81% of today’s accepted value of 186,282 mi/s.BFor: Information about the speed of lightWeb Code: aue-2032Lesson 7-1 Ratios and Proportions57. ba dc (Given); ad bc(Cross-Prod. Prop.);bcadcd cd (Div. Prop.bac d (Simplify.)yAIn 1675, Danish astronomer Ole Römer56. ba dc (Given); ad bc(Cross-Product Prop.);bc ad (Symm. Prop.adof ); bcac ac (Div. Prop.bdof ); a c (Simplify.)72.B2000PHSchool.comrectangle, parallelogramof );37158. ba dc (Given); ba 1 cd 1 (Add. Prop. of );dabcb b d d (Subst.);a 1 bd c1bd (Simplify.)x4rhombus, parallelogram75. a. If an l is not acute,then it does not havemeasure between 0and 90.b. If an l does not havemeasure between 0and 90, then it is notacute.76. a. If two lines are not n,then they are notcoplanar.b. If two lines are notcoplanar, then theyare not n.77. a. If two ' are notcompl., then the 'are not both acute.b. If two ' are not bothacute, then they arenot compl.371

Lesson 7-1 Ratios and Proportions 367 A is a statement that two ratios are equal.You can write a proportion in these forms: and a b c d When three or more ratios are equal, you can write an For example, you could write the following: Two equations are equivalent when either can be

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