Geom 3eTE.1004.X 553-558 - Portal.mywccc
10-4Perimeters and Areas ofSimilar Figures10-41. PlanGO for HelpWhat You’ll LearnCheck Skills You’ll Need To find the perimeters andFind the perimeter and area of each figure.areas of similar figures1. . . And WhyTo find the expected yield ofa garden, as in Example 32.3.Lesson 1-924 cm; 24 cm24m5. b 2 cm, h 6 cm16 cm; 12 cm21To find the perimeters andareas of similar figuresExamples6 cm8m7 in.8 cm28 in.; 49 in.224 m; 32 m2Find the perimeter and area of each rectangle with the given base and height.4. b 1 cm, h 3 cm8 cm; 3 cm2Objectives6. b 3 cm, h 9 cm24 cm; 27 cm2123Finding Ratios in SimilarFiguresFinding Areas Using SimilarFiguresFinding Similarity andPerimeter RatiosMath Background1Finding Perimeters and Areas of Similar FiguresHands-On Activity: Perimeters and Areas of Similar Rectangles On a piece of grid paper, draw a 3-unit by 4-unit rectangle. Draw three different rectangles, each similar to the original rectangle.Label them I, II, and III.1. Use your drawings to complete a chart like this. Check students’ work.RectangleOriginalIIIIIIPerimeterAreaThe Distributive Property readilyproves that the ratio of theperimeters of two similar figureswith the similarity ratio a : b isalso a : b. To prove that the ratioof the areas of two similartriangles with the similarity ratioa : b is a2 : b2, draw altitudes tocorresponding sides and provethat the right triangles thusformed are similar. The TransitiveProperty allows the proportionalrelationship of the triangles’ sidesto be extended to their altitudes.More Math Background: p. 530CLesson Planning andResources2. Use the information from the first chart to complete a chart like this.Check students’ work.SimilarityRatioRectangleI to OriginalII to OriginalIII to OriginalRatio ofPerimetersSee p. 530E for a list of theresources that support this lesson.Ratio ofAreasPowerPointBell Ringer PracticeCheck Skills You’ll Need3. The ratio for perimetersis the same, but theratio for areas is thesimilarity ratio squared.3. How do the ratios of perimeters and the ratios of areas compare withthe similarity ratios? See left.Lesson 10-4 Perimeters and Areas of Similar Figures553For intervention, direct students to:Finding PerimeterLesson 1-9: Examples 1 and 2Extra Skills, Word Problems, ProofPractice, Ch. 1Finding AreaSpecial NeedsBelow LevelL1For the Hands-On Activity, have students usegeoboards to create the similar rectangles. Havestudents start with a 2-unit by 3-unit rectangle, andrestrict the choice of scale factors to fit on ageoboard.learning style: tactileL2Before you go over Theorem 10-7, have students drawa triangle and three midsegments. Discuss how thefour congruent triangles relate to Theorem 10-7.learning style: visualLesson 1-9: Example 4Extra Skills, Word Problems, ProofPractice, Ch. 1553
2. TeachTo compare areas of similar figures, you can square the similarity ratio.Key ConceptsTheorem 10-7If the similarity ratio of two similar figures is ba , then(1) the ratio of their perimeters is ba andGuided Activity2(2) the ratio of their areas is a2.Hands-On ActivitybBecause all rectangles have fourright angles, remind students thatall rectangles with a 3 : 4 ratioof sides are similar.1EXAMPLE1Math TipEXAMPLEFinding Ratios in Similar FiguresEXAMPLEThe trapezoids at the right are similar. The ratioof the lengths of corresponding sides is 69 , or 23 .Point out that the ratios of theperimeters and areas were foundwithout calculating the perimeteror area of either trapezoid. Infact, those measurements cannotbe found for the given figuresbecause only one side length ofeach is known.2Perimeters and Areas of Similar Figures6ma. Find the ratio (smaller to larger) of the perimeters.The ratio of the perimeters is the same as the ratioof corresponding sides, which is 23.9mb. Find the ratio (smaller to larger) of the areas.The ratio of the areas is the square of the ratio2of corresponding sides, which is 22, or 49.3Error PreventionQuick CheckRemind students not to use thesimilarity ratio as the ratio ofthe areas. Point out that area ismeasured in square units, so theratio of the areas is the squareof the similarity ratio.1 Two similar polygons have corresponding sides in the ratio 5 ; 7.a. Find the ratio of their perimeters. 5 : 7b. Find the ratio of their areas. 25 : 49When you know the area of one of two similar polygons, you can use a proportionto find the area of the other polygon.Teaching TipAfter students finish Example 2,ask: How do you know that allregular pentagons are similar?All regular figures are equilateraland equiangular. So, all angles ofregular pentagons are congruent,and the ratio of the sides ofany two regular pentagons isconstant.3EXAMPLEVisual LearnersHave students draw a rectanglefor each plot of land to help themvisualize the descriptions.4EXAMPLE2Multiple Choice The area of the smaller regular pentagonis about 27.5 cm 2. What is the best approximation for thearea of the larger regular king TipYou can eliminatechoice A immediately.The area of the largerpentagon must begreater than 27.5 cm2.11 cm2172 cm269 cm2275 cm227.5425 A10 cm5A 687.54 171.875Write a proportion.Cross-Product PropertySolve for A.The area of the larger pentagon is about 172 cm 2. The answer is C.Quick Check5542 The corresponding sides of two similar parallelograms are in the ratio 3.4The area of the larger parallelogram is 96 in.2. Find the area of thesmaller parallelogram. 54 in.2Chapter 10 AreaAdvanced LearnersEnglish Language Learners ELLL4After Example 2, have students prove that the ratio ofthe areas of two similar regular polygons equals thesquare of the ratios of their sides.5544 cmRegular pentagons are similar because all angles measure 108 and all sides in each4are congruent. Here the ratio of corresponding-side lengths is 10, or 25. The ratio224of the areas is 2, or 25.4A 687.5Connectionto AlgebraStudents are used to solving anequation for one variable butnot for the ratio of two variables.Discuss why taking the squareroot of a ratio is like solvingtwo equations.Finding Areas Using Similar FiguresEXAMPLElearning style: verbalSome students may confuse the ratio of perimeters ofsimilar figures with the ratio of areas of similarfigures. Point out that perimeter is a linear measurewhile area is measured in square units, and itssimilarity ratio is squared.learning style: verbal
PowerPoint3Real-WorldEXAMPLEConnectionAdditional ExamplesCommunity Service During the summer, a group of high school students used aplot of city land and harvested 13 bushels of vegetables that they gave to a foodpantry. Their project was so successful that next summer the city will let them use alarger, similar plot of land.In the new plot, each dimension is 2.5 times the corresponding dimension of theoriginal plot. How many bushels can they expect to harvest next year?Real-WorldConnectionThe ratio of the dimensions is 2.5 ; 1. So, the ratio of the areas is (2.5)2 ; 12, or6.25 ; 1. With 6.25 times as much land next year, the students can expect to harvest6.25(13), or about 81 bushels.Many cities make city landavailable to the communityfor gardening.Quick Check3 The similarity ratio of the dimensions of two similar pieces of window glass is 3 ; 5.The smaller piece costs 2.50. What should be the cost of the larger piece? 6.94When you know the ratio of the areas of two similar figures, you can workbackward to find the ratio of their perimeters.4Find the similarity ratio a ; b.For: Perimeter and Area ActivityUse: Interactive Textbook, 10-4Quick Check6556.257.5perimeters:54;areas: 25162 The ratio of the lengths of thecorresponding sides of two regularoctagons is 83. The area of the largeroctagon is 320 ft2. Find the area ofthe smaller octagon. 45 ft2two rectangular fields, each withside lengths in a ratio of 2 : 3. Eachdimension of the larger field is 312times the dimension of the smallerfield. Seeding the smaller fieldcosts 8. How much money doesseeding the larger field cost? 98The areas of two similar triangles are 50 cm2 and 98 cm2. What is the similarityratio? What is the ratio of their perimeters?a 2 5098b2a 2 2549b2a 57b43 Benita plants the same crop inFinding Similarity and Perimeter RatiosEXAMPLE1 The triangles below are similar.Find the ratio (larger to smaller) oftheir perimeters and of their areas.The ratio of the areas is a 2 ; b 2.4 The areas of two similarpentagons are 32 in.2 and 72 in.2What is their similarity ratio? Whatis the ratio of their perimeters?2 : 3; 2 : 3Simplify.Take square roots.The ratio of the perimeters equals the similarity ratio 5 ; 7.4 The areas of two similar rectangles are 1875 ft 2 and 135 ft 2. Find the ratio of theirperimeters. 5 "5 : 3Resources Daily Notetaking Guide 10-4L3 Daily Notetaking Guide 10-4—L1Adapted InstructionEXERCISESFor more exercises, see Extra Skill, Word Problem, and Proof Practice.Practice and Problem SolvingAClosurePractice by ExampleExample 1GO forHelp(page 554)The figures in each pair are similar. Compare the first figure to the second. Give theratio of the perimeters and the ratio of the areas.4 : 3; 16 : 91.2.1 : 2; 1 : 48 cm2 in.6 cm4 in.3.The similarity ratio of two similartriangles is 5 : 3. The perimeterof the smaller triangle is 36 cm,and its area is 18 cm2. Find theperimeter and area of the largertriangle. perimeter: 60 cm;area: 50 cm24.14 m2 : 3; 4 : 921 m15 in.25 in.3 : 5; 9 : 25Lesson 10-4 Perimeters and Areas of Similar Figures555555
3. PracticeExample 2(page 554)The figures in each pair are similar. The area of one figure is given. Find the area ofthe other figure to the nearest whole number.5.Assignment Guide6. 54 m224 in.21 A B 1-4012 mC Challenge41-44Test PrepMixed Review45-4950-613 in.6 in.Area of smaller parallelogram 6 in.259 ft27.16 ftHomework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 4, 12, 35, 38, 39.8.439 m212 ft5m13 mArea of smaller hexagon 65 m2Example 3Exercise 23 Misleading graphs(page 555)often are found in magazines andnewspapers, so everyone needs toknow how to analyze graphscritically. Have students suggesthow they would draw a moreappropriate graph.9. Remodeling It costs a family 216 to have a 9 ft-by-12 ft wooden floorrefinished. At that rate, how much would it cost them to have a 12 ft-by-16 ftwooden floor refinished? 38410. Decorating An embroidered placemat costs 2.95. An embroidered tableclothis similar to the placemat, but four times as long and four times as wide. Howmuch would you expect to pay for the tablecloth? 47. 20Example 4Exercise 22 Watch for students(page 555)2who use a ratio of s 2 instead of4s(4s) 2Area of larger trapezoid 121 m 2Area of larger triangle 105 ft2Connection to Statisticss218 m25 s 2. Ask: If a side is four16stimes larger, how much largerwould its area be? 42 16 timeslargerBApply Your SkillsFind the similarity ratio and the ratio of perimeters for each pair of similar figures.11. two regular octagons1 : 2; 1 : 2with areas 4 ft2 and 16 ft212. two triangles5 : 2; 5 : 2with areas 75 m 2 and 12 m 213. two trapezoids7 : 3; 7 : 3with areas 49 cm 2 and 9 cm 214. two parallelograms 3 : 4; 3 : 4with areas 18 in.2 and 32 in.215. two equilateral triangles 4 : 1; 4 : 1with areas 16 !3 ft 2 and !3 ft 216. two circles1 : 10; 1 : 10with areas 2p cm 2 and 200p cm 2The similarity ratio of two similar polygons is given. Find the ratio of theirperimeters and the ratio of their areas.17. 3 ; 118. 2 ; 519. 2320. 7421. 6 ; 12 : 5; 4 : 253 : 1; 9 : 16 : 1; 36 : 17 : 4; 49 : 162 : 3; 4 : 922. Multiple Choice The area of a regular decagon is 50 cm2. What is the area of aregular decagon with sides four times the sides of the smaller decagon? C200 cm2GPS Guided Problem SolvingL3L4EnrichmentL2ReteachingL1Adapted PracticePracticeNameClassL3DatePractice 1-1GOnline1. 17, 23, 29, 35, 41, c2. 1.01, 1.001, 1.0001, c3. 12, 14, 18, 24, 32, c4. 2, 4, 8, 16, 32, c5. 1, 2, 4, 7, 11, 16, c6. 32, 48, 56, 60, 62, 63, cThenName two different ways to continue each pattern.7. 1, 1, 2, 98. 48, 49, 50, 910. A, B, C, c, Z, 99. 2, 4, 911. D, E, F, 912. A, Z, B, 9Draw the next figure in each sequence.13.?14.556?15. Pearson Education, Inc. All rights reserved.90 135 157.5 Seven people meet and shake hands with one another.16. How many handshakes occur?17. Using inductive reasoning, write a formula for the number of handshakes if thenumber of people is n.The Fibonacci sequence consists of the pattern 1, 1, 2, 3, 5, 8, 13, . . .18. What is the ninth term in the pattern?19. Using your calculator, look at the successive ratios of one term to the next.Make a conjecture.20. List the first eight terms of the sequence formed by finding the differences ofsuccessive terms in the Fibonacci sequence.556?NowVisit: PHSchool.comWeb Code: aue-1004Patterns and Inductive ReasoningChapter 10 Area800 cm22000 cm223. Error Analysis A reporter used the graphic below to show that the number ofhouses with more than two televisions had doubled in the past few years.Explain why this graphic is misleading.While the ratio of lengths is2 : 1, the ratio of areas is 4 : 1.Homework HelpFind a pattern for each sequence. Use the pattern to show the nexttwo terms.500 cm2
24. Medicine For some medical imaging, the scale of the image is 3 i 1. That meansthat if an image is 3 cm long, the corresponding length on the person’s body is1 cm. Find the actual area of a lesion if its image has area 2.7 cm2. 0.3 cm225. The longer sides of a parallelogram are 5 m. The longer sides of a similarparallelogram are 15 m. The area of the smaller parallelogram is 28 m2. What isthe area of the larger parallelogram? 252 m22x Algebra Find the values of x and y when the smaller triangle shown here has thegiven area.8 cm27. 6 cm228. 12 cm226. 3 cm2x26–31.29. 16 cm230. 24 cm231. 48 cm2 See margin. y12 cmReal-WorldConnectionCareers Doctors use enlargedimages to aid in certainmedical procedures.Problem Solving HintFor Exercise 34, recallthe length of adiagonal of a squarewith 2-in. sides.32. Two similar rectangles have areas 27 in.2 and 48 in.2. The length of one side ofthe larger rectangle is 16 in. What are the dimensions of both rectangles?214 in. by 12 in.,33. In #RST, RS 20 m, ST 25 m, and RT 40 m.3 in. by 16 in.a. Open-Ended Choose a convenient scale. Then use a ruler and compass todraw #R9S9T9 , #RST. Check students’ work.b. Constructions Construct an altitude of #R9S9T9 and measure its length.Find the area of #R9S9T9. Check students’ work.c. Estimation Estimate the area of #RST. Estimates may vary. Sample: 205 m234. Drawing Draw a square with an area of 8 in.2. Draw a second square with anarea that is four times as large. What is the ratio of their perimeters?Ratio of small to large is 1 : 2.Compare the blue figure to the red figure. Find the ratios of (a) their perimetersand (b) their areas.35.36.8 643; 937.2 41; 13 cm5 252; 48 cm38. Answers may vary.Sample: The proposedplayground is more thanadequate. The numberof students hasapproximately doubled.The proposedplayground would befour times larger thanthe original playground.39b. 114 mm; 475 mm238. Writing The enrollment at an elementary school is going to increase from200 students to 395 students. A parents’ group is planning to increase the100 ft-by-200 ft playground area to a larger area that is 200 ft by 400 ft. Whatwould you tell the parents’ group when they ask your opinion about whetherthe new playground will be large enough? See left.39. a. Surveying A surveyor measured one sideGPS and two angles of a field as shown in thediagram. Use a ruler and a protractor todraw a similar triangle. See margin.b. Measure the sides and altitude of yourtriangle and find its perimeter and area.c. Estimation Estimate the perimeter andarea of the field. 456 yd; 7600 yd230 26. x 2 cm, y 3 cm27. x 2 "2 cm,y 3 "2 cm50 29. x 8Á3 3 cm,y 4 "3 cm1. For the similar rectangles, givethe ratios (smaller to larger) ofthe perimeters and of theareas.4 cm9 cmperimeters:49;areas: 16812. The triangles below are similar.The area of the larger triangleis 48 ft2. Find the area of thesmaller triangle.8 ft6 ft27 ft24. The areas of two equilateraltriangles are 27 yd2 and75 yd2. Find their similarityratio and the ratio of theirperimeters. 3 : 5; 3 : 55. Mulch to cover an 8-ft by 16-ftrectangular garden costs 48.At the same rate, what wouldbe the cost of mulch to cover a12-ft by 24-ft rectangulargarden? 108200 yd6 cm3 cm8 cmLesson 10-4 Perimeters and Areas of Similar Figures28. x 4 cm, y 6 cmLesson QuizAlternative Assessment40. a. Find the area of a regular hexagon withsides 2 cm long. Leave your answer insimplest radical form. 6 "3 cm 2b. Use your answer to part (a) andTheorem 10-7 to find the areas of the regularpolygons shown at the right.54 "3 cm 2; 13.5 "3 cm 2; 96 "3 cm 2lesson quiz, PHSchool.com, Web Code: aua-1004PowerPoint3. The similarity ratio of tworegular octagons is 5 : 9. Thearea of the smaller octagon is100 in.2 Find the area of thelarger octagon. 324 in.22x5x4. Assess & Reteach30. x 4 "2 cm,y 6 "2 cm31. x 8 cm,y 12 cm557Have students work in pairs anduse rulers and graph paper toestimate the area of a map ofyour state. Then have them usethe map scale and Theorem 8-6to estimate the actual area ofthe state.39. Answers may vary.Sample:a.39 mm25 mm19 mm50 mm557
2. TeachWhen you substitute x 0 into the equation y ax 2 bx c, y c. So they-intercept of a quadratic function is the value of c. You can use the axis ofsymmetry and the y-intercept to help you graph a quadratic function.Guided Instruction1EXAMPLE1Math TipStep 1 Find the equation of the axis of symmetry and the coordinates ofthe vertex.26x 2b2a 2(23) 1 Find the equation of the axis of symmetry.The axis of symmetry is x 1.For: Quadratic Function ActivityUse: Interactive Textbook, 10-2Additional Examplesy 4x - 3.Step 2 Find two other points on the graph.Use the y-intercept.x 2Oy -3(1)2 6(1) 5 To find the y-coordinate of the vertex, substitute 1 for x.The vertex is (1, 8).y 4y -3x 2 6x 5 81 Graph the function2x2Graphing y ax 2 bx cGraph the function y -3x 2 6x 5.Encourage students to enterY1 -3x 2 6x 5 into theirgraphing calculators. Have thempress 2nd TABLE and identify pairsof points which are reflectionsacross the axis of symmetry.PowerPointEXAMPLEFor x 0, y 5, so one point is (0, 5).2 2Choose a value for x on the same side of the vertex as the y-intercept. 4Let x -1.y -3(-1)2 6(-1) 5 Find the y-coordinate for x –1. -42 Suppose a particular star isprojected from an aerial fireworkat a starting height of 610 ftwith an initial upward velocityof 88 ft/s. How long will it takefor the star to reach its maximumheight? How far above the groundwill it be? 2.75 s; 731 ftFor x -1, y -4, so another point is (-1, -4).Step 3 Reflect (0, 5) and (-1, -4) across the axis of symmetry to get twomore points. Then draw the parabola.y(0, 5)1.yx 3x 1y(1, 8)(2, 5)5x 14xO 2xO242O2(3, 0)4 xQuick Check( 1, 4)(3, 4)1 Graph f(x) x 2 - 6x 9. Label the axis of symmetry and the vertex.See left.You saw in the previous lesson that the formula h -16t 2 c describes theheight above the ground of an object falling from an initial height c, at time t.If an object is given an initial upward velocity v and continues with no additionalforce of its own, the formula h -16t 2 vt c describes its approximate heightabove the ground.558Chapter 10 Quadratic Equations and FunctionsAdvanced LearnersEnglish Language LearnersL4Have students graph the quadratic function inExample 2.558learning style: verbalELLAsk students if they have any ideas about how to findthe area under a curve, as in Exercises 38 and 39.Explain that if no formula comes to mind, estimationis good problem-solving strategy for finding theanswer to some questions.learning style: verbal
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 .
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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
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 .
TruVision DVR 10 User Manual 1 Chapter 1 Product introduction Product overview This is the TruVision DVR 10 User Manual for models: TVR-1004-250 TVR-1004-500 TVR-1004-1T The TruVision DVR 10 (TVR 10) is a network digital video recorder developed for digital surveillance. The TVR 10 uses an embedded microcontroller unit with the Linux
Financial Accounting Working Papers, Robert F. Meigs, Jan R. Williams, Sue Haka, Susan F. Haka, Mark S Bettner, Jun 1, 2000, Business & Economics, 400 pages. . Accounting Chapters 1-14 The Basis for Business Decisions, Robert F. Meigs, Jan R. Williams, Sue Haka, Susan F. Haka, Mark S. Bettner, Sep 1, 1998, Business & Economics, . The Study Guide enables the students to measure their progress .
