Area CHAPTER. Overview. The Mathematics. Area 421A

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1 CHAPTER 8 Are Overview In Lessons 8.1, 8., 8.4, nd 8.5, students discover formuls for finding te res of rectngles, prllelogrms, tringles, trpezoids, kites, regulr polygons, nd circles y cutting prt nd rerrnging tese figures to derive new formuls from previously discovered ones. Beyond simply plugging numers into formuls, students sould e le to descrie te process y wic tey rrived t te formuls. Advnced students sould lso e le to demonstrte te lger tt supports te discoveries. Prolem solving is empsized in Lesson 8.3, were te prolems my require severl steps. Be prepred to offer etr elp to students wose Englis skills re limited, ut don t sy wy from ssigning te prolems. In Using Your Alger Skills 8, students use rectngle digrms to review fctoring nd pply te zero product property to solve qudrtic equtions. After Lesson 8.4, te eplortion on Pick s formul introduces wy to estimte re. Students comine formuls from erlier lessons s tey eplore res of prts of circles in Lesson 8.6 nd surfce re in Lesson 8.7. An eplortion on proility precedes Lesson 8.7, nd Sketcpd eplortion emining oter wys of clculting re concludes te cpter. Te Mtemtics Are Some students will sy, O, re. Tt s lw. An empsis on memorizing re formuls my ve misled your students into tinking tey understnd re. Te mening lies not in wt te formuls re ut rter in understnding wy te formuls re wt tey re. You cn open your students minds to ides out re tt go eyond formuls tey my lredy ve seen. Here re some emples: Te sttement A rectngle does not ve re is true. Te rectngle consists of four line segments, ec wit re zero. In generl, none of te spes we ve een studying include teir interiors. So te prse te re of rectngle is sortnd for te re of te region ounded y te rectngle. Cn figure ve more tn one re? A quick response migt e Of course not. But isn t te re of figure in squre centimeters different from its re in squre yrds? On te oter nd, if spe cn ve different res, wy do we instinctively feel tt it s only one? Could it e tt we re using te term re in different sense, to men region rter tn numer tt denotes te mesure of region? Wt ppens to formuls if te mesurement units re converted to oter units? Does te formul for te re of trpezoid, memorized wit gret effort, continue to pply wen we cnge te lengt mesurements from centimeters to miles? If so, wy? Students my not ve deep understnding of tis ide. Even te ncient Greeks knew tt te rtio of te circumference of circle to its dimeter ws constnt, te vlue we now cll. But tey did not relize tt lso ppers in formul for te re of circle. Wy does it? An strct definition of re, eyond te understnding of most students t vn Hiele level, is s function tt ssigns to ec two-dimensionl geometric spe nonnegtive rel numer so tt (1) te re of every point is zero, () te res of congruent figures re equl, nd (3) if spe is prtitioned into suregions, ten te sum of te res of tose suregions equls te re of te spe. Tere re infinitely mny suc functions, so spes cn ve infinitely mny res. Despite its strction, tis definition points to two oter ides out res tt migt surprise some of your students: If tringle or prllelogrm is rotted so tt different side is considered te se, does te re formul necessrily give te sme result? In generl, under wt trnsformtions is te re conserved? Some students my not know. CHAPTER 8 INTERLEAF 41A

2 Wt if rectngle is cut into pieces tt re ten rerrnged to mke, sy, prllelogrm? Will te re relly e te sme? If you puse to rise te question, you my find tt your students ve lot of dout. Surfce Are Just s rectngle doesn t include te spce it encompsses, geometric solid doesn t include its interior. We cn tlk out te volume of solid s we will in Cpter 10 ut we won t men te volume of te solid itself ut rter te volume of te region enclosed y te solid. In most uses of te term solid, its oundry (te interiors of te polygons formed y its edges) is prt of te solid. Surfce re is te re of tt oundry. Help students lern te terminology of solids. Te term sides pplies to polygons. For solids, refer to fces nd edges s well s vertices. Using Tis Cpter Ner te end of te cpter, ssign nd discuss Improving Your Visul Tinking Skills: Cover te Squre s preprtion for Cpter 9. Pln to use two dys for te cpter review, perps doing te eplortions in clss on one of tose dys nd sring results nd questions out te review on te oter dy. Resources Discovering Geometry Resources Tecing nd Workseet Msters Lessons , 8.6, 8.7 Eplortion: Pick s Formul for Are Discovering Geometry wit Te Geometer s Sketcpd Lessons 8.1, 8., 8.4, 8.5 Assessment Resources Quiz 1 (Lessons 8.1, 8.) Quiz (Lessons ) Quiz 3 (Lessons 8.6, 8.7) Cpter 8 Test Cpter 8 Constructive Assessment Options Prctice Your Skills for Cpter 8 Condensed Lessons for Cpter 8 Oter Resources Tecing Mtemtics wit Ftom from Key Curriculum Press. For complete references to tis nd oter resources, see flourisk.com. Mterils construction tools scissors evy pper or crdord grp pper geoords, optionl rulers clcultors lger tiles, optionl squre dot pper, optionl tpe or removle glue stick, optionl pennies nd dimes protrctors prisms, pyrmids, nd oter solids Te Geometer s Sketcpd, optionl 41B CHAPTER 8 INTERLEAF Are

3 CHAPTER 8 Are CHAPTER 8 OBJECTIVES I could fill n entire second life wit working on my prints. M. C. ESCHER Squre Limit, M. C. Escer, Cordon Art B. V. Brn Hollnd. All rigts reserved. In Squre Limit, Escer refers to geometry in wic te eternl nd internl mesurements of size differ. Viewed eternlly, te res of te fis cnge s tey move wy from te center. But so would te lengts of ny rulers tey were crrying. Terefore te fis temselves internlly could detect no cnge in teir sizes. Tere is muc to e lerned out size (re) eyond formuls. OBJECTIVES In tis cpter you will discover re formuls for rectngles, prllelogrms, tringles, trpezoids, kites, regulr polygons, circles, nd oter spes use re formuls to solve prolems lern ow to find te surfce res of prisms, pyrmids, cylinders, nd cones [Ask] At te center is squre consisting of fins from four fis. Were nd ow is tt squre repeted? [long te digonls wit one wite fin, two red (or green) fins, nd one green (or red) fin] Wy do you tink te rtist cose te title Squre Limit? [Te rtwork is squre; te size of te fis pproces limit of zero ner te outside edges.] [Link] Limit is fundmentl concept in clculus. Develop te concept of re Derive formuls nd metods for finding te res of rectngles, prllelogrms, tringles, trpezoids,kites,regulr polygons, circles, sectors, segments, nd nnuluses Use rectngle digrms to review fctoring Discover Pick s formul for finding te re of n irregulr figure drwn on grid Discover metods for finding te surfce res of solids Lern new voculry Solve re ppliction prolems using vrious prolem-solving strtegies Review terminology for solids Clrify pproimtion concepts Prctice visul tinking in tree dimensions Develop reding compreension nd coopertive evior Prctice using geometry tools Prctice mesuring Prctice estimtion CHAPTER 8 Are 41

4 LESSON 8.1 PLANNING LESSON OUTLINE One dy: 10 min Investigtion 5min Emples 15 min Sring 15 min Closing nd Eercises MATERIALS construction tools scissors evy pper grp pper, optionl geoords, optionl Prllelogrm (W) for One Step Sketcpd ctivity Ares of Rectngles nd Prllelogrms, optionl LESSON 8.1 A little lerning is dngerous ting lmost s dngerous s lot of ignornce. ANONYMOUS Ares of Rectngles nd Prllelogrms People work wit res in mny occuptions. Crpenters clculte te res of wlls, floors, nd roofs efore tey purcse mterils for construction. Pinters clculte surfce res so tt tey know ow muc pint to uy for jo. Decortors clculte te res of floors nd windows to know ow muc crpeting nd drpery tey will need. In tis cpter you will discover formuls for finding te res of te regions witin tringles, prllelogrms, trpezoids, kites, regulr polygons, nd circles. Te re of plne figure is te mesure of te region enclosed y te figure. You mesure te re of figure y counting te numer of squre units tt you cn rrnge to fill te figure completely. 1 Lengt: 1 unit 1 1 Are: 1 squre unit Tile lyers need to find floor re to determine ow mny tiles to uy. TEACHING In tis lesson you cn deepen students understnding of te re of figure wile tey encounter some formuls tt migt lredy e fmilir to tem. [ELL] Strt voculry disply tt will e used trougout tis cpter. At tis point te disply sould include te following terms wit digrms: rectngle, prllelogrm, se, eigt, re nd region. One Step Hnd out copy of te Prllelogrm workseet to ec student. Cllenge groups to find te re of te figure in s mny wys s tey cn. Some students my rememer formul ut ve difficulty deciding wic edge sould e te se. Oters my see te figure s union of tringles or s te result of removing tringles from rectngle. Oters my copy it onto grp pper nd count squres. Oters my cut it up nd rerrnge pieces. If some of tese ides ren t emerging from ny group, you migt plnt te ides wen you visit groups tt re ed of te oters. During Sring, [Ask] Sould ll te nswers e te sme? [Answers migt differ due to te inectness in counting or mesuring. Tey will lso differ if different mesurement units re used.] You proly lredy know mny re formuls. Tink of te investigtions in tis cpter s pysicl demonstrtions of te formuls tt will elp you understnd nd rememer tem. It s esy to find te re of rectngle. To find te re of te first rectngle, you cn simply count squres. To find te res of te oter rectngles, you could drw in te lines nd count te squres, ut tere s n esier metod. INTRODUCTION Ask wt students know out re. Mny will sy, Are is lw. Bring out te ide tt tey my e tinking of formul for clculting te re of rectngle. Discuss te ide tt re often mens numer ssocited wit te region enclosed y te spe. Going ck to te fmilir formul, [Ask] Wt do l nd w stnd for? Ten sk students ow tey figure out wic side is te lengt nd wic is te widt. 4 CHAPTER 8 Are

5 EXAMPLE A Solution Any side of rectngle cn e clled se. A rectngle s eigt is te lengt of te side tt is perpendiculr to te se. For ec Heigt pir of prllel ses, tere is Heigt corresponding eigt. Bse Bse If we cll te ottom side of ec rectngle in te figure te se, ten te lengt of te se is te numer of squres in ec row nd te eigt is te numer of rows. So you cn use tese terms to stte formul for te re. Add tis conjecture to your list. Rectngle Are Conjecture A Te re of rectngle is given y te formul?,were A is te re, is te lengt of te se, nd is te eigt of te rectngle. Te re formul for rectngles cn elp you find te res of mny oter spes. Find te re of tis spe. Te middle section is rectngle wit n re of 4 8, or 3 squre units. You cn divide te remining pieces into rigt tringles, so ec piece is ctully lf rectngle. Te re of te figure is squre units. Tere re oter wys to find te re of tis figure. One wy is to find te re of n 8-y-9 rectngle nd sutrct te res of four rigt tringles. 1 1 (6) 3 (10) 5 3 C-74 [Lnguge] Te dictionry defines lengt s te mesure of te gretest dimension of plne or solid figure. Widt is often used for te sorter of te two sides. Becuse te re of rectngle is te product of two djcent sides lengts, tis ook uses se nd eigt to eliminte ssociting term wit prticulr size. [Lnguge] Bse refers ot to te segment nd to its mesure. Cution students tt se segment migt not e orizontl; in fct, te figure migt e drwn wit no orizontl sides. Te numer of squres does not ve to e n integer. You migt cllenge students to drw on grp pper rectngle wose sides don t ve integer lengts nd to see ow te frctionl prts ccumulte wit te wole numers wen te rows re counted. EXAMPLE A Encourge students to estimte te re efore reding te solution. [Ask] Wt would te re of te spe e if te ottom verte were moved two units to te rigt? [Te re remins te sme, 5 squre units.] 1 (6) 3 1 (18) 9 LESSON OBJECTIVES Develop te concept of re Derive formuls for te res of rectngle nd prllelogrm Apply re formuls to solve prolems Review voculry LESSON 8.1 Ares of Rectngles nd Prllelogrms 43

6 Guiding te Investigtion You migt use te investigtion s follow-long ctivity or ve student demonstrte it to te clss. You migt lso wis to etend or replce tis investigtion wit te Dynmic Geometry Eplortion t flourisk.com. Step 1 As needed, remind students tt n ltitude of prllelogrm is perpendiculr segment from one side to nondjcent side (or to line contining te side). Step If students cut long perpendiculr tt s not troug verte, sk tem to present teir pproc for critique during Sring. SHARING IDEAS As students present teir ides, sk wy te prllelogrm nd te rectngle ve te sme re. Te re of ec is te sum of te res of its prts. (You migt mention tt tis is often clled te Are Addition Postulte. It is te sis of most of te informl derivtions of re formuls in te cpter.) [Ask] Is tere more tn one wy to cut up te prllelogrm so tt its pieces cn e rerrnged into rectngle? [Cut troug te prllelogrm long ny ltitude from one se to te oter.] Ask wic of te sides of te prllelogrm could e tken to e te se. Drw n ltitude to noter side nd sk weter te product of tt se nd eigt will e te sme s te product of te originl se nd eigt. You need not nswer tis question now; it cn e revisited in Cpter 9. Ask weter it s esier to count squres to find te re of te rectngle tn it is to count squres of te prllelogrm. Some students my tink so, ut You will need evy pper or crdord strigtedge compss Step 1 Step yes, ecuse Step no re ws lost or gined Step 3 You cn lso use te re formul for rectngle to find te re formul for prllelogrm. Just s wit rectngle, ny side of prllelogrm cn e clled se. But te eigt of prllelogrm is not necessrily te lengt of side. An ltitude is ny segment from one side of prllelogrm perpendiculr to line troug te opposite side. Te lengt of te ltitude is te eigt. Bse Altitude to te se Te ltitude cn e inside or outside te prllelogrm. No mtter were you drw te ltitude to se, its eigt sould e te sme, ecuse te opposite sides re prllel. let tem see tt, s in rectngle, ec row of te prllelogrm s te sme numer of squres. In fct, tey my see tt tose rows cn simply e sifted nd lined up to mke te corresponding rows of te rectngle. Bring up te ide of mesurement units. [Ask] Wen we sy tt prllelogrm s re ninety, wt re te units? [It depends on te units of lengt y wic te liner dimensions re mesured. If te se nd eigt re in cm, ten te re is in cm ;90 cm mens tt 90 squres 1 cm Investigtion Are Formul for Prllelogrms Using evy pper, investigte te re of prllelogrm. Cn te re e rerrnged into more fmilir spe? Different memers of your group sould investigte different prllelogrms so you cn e sure your formul works for ll prllelogrms. Construct prllelogrm on piece of evy pper or crdord. From te verte of te otuse ngle djcent to te se, drw n s s ltitude to te side opposite te se. Lel te prllelogrm s sown. Bse Altitude to te se Cut out te prllelogrm nd ten cut long te ltitude. You will ve two pieces tringle nd trpezoid. Try rrnging te two pieces into oter spes witout overlpping tem. Is te re of ec of tese new spes te sme s te re of te originl prllelogrm? Wy? Is one of your new spes rectngle? Clculte te re of tis rectngle. Wt is te re of te originl prllelogrm? Stte your net conjecture. Prllelogrm Are Conjecture A Te re of prllelogrm is given y te formul?,were A is te re, is te lengt of te se, nd is te eigt of te prllelogrm. or C-75 on side could fit into te prllelogrm, wit cutting nd rerrnging.] Wt s te difference etween 90 squre centimeters nd 90 centimeters squred? [Te former refers to 90 squres, ec 1 cm on side. Te ltter is te squre of 90 cm, or 8100 cm.] Te eponent on te unit cm refers to te cm, not to te numer. To reduce confusion, te nottion cm sould e red squre centimeters, not centimeters squred. [Ask] How re te prllelogrm nd te rectngle te sme, nd ow re tey different? [One pir of 44 CHAPTER 8 Are

7 flourisk.com EXAMPLE B Solution EXERCISES [ To view dynmiclly ow te re of prllelogrm is relted to te re of rectngle, see te Dynmic Geometry Eplortion Are Formul for Prllelogrms t flourisk.com. ] If te dimensions of figure re mesured in inces, feet, or yrds, te re is mesured in in (squre inces), ft (squre feet), or yd (squre yrds). If te dimensions re mesured in centimeters or meters, te re is mesured in cm (squre centimeters) or m (squre meters). Let s look t n emple. Find te eigt of prllelogrm tt s re 7.13 m nd se lengt.3 m. A Write te formul (.3) Sustitute te known vlues Solve for te eigt. 3.1 Divide. Te eigt mesures 3.1 m. In Eercises 1 6, ec qudrilterl is rectngle. A represents re nd P represents perimeter. Use te pproprite unit in ec nswer. 1. A? 8 m. A? cm 3. A 96 yd? 8 yd 1 m 4.5 cm 1 yd 9.3 cm 19 m 4. A 73 cm 5. P 40 ft 6. Sded re? 18 m? 1 cm A? 91 ft 13 cm In Eercises 7 9, ec qudrilterl is prllelogrm. 7. A? 96 in 8. A 508 cm 9. Find te re of te P? 10 cm sded region. A 4 ft 9 in. 8 in. 1 in. 7 ft 44 cm EXAMPLE B If confusion rises over te mesurement units, point out tt 7.13 m is not 7.13 meters squred, ut 7.13 squre meters. Dimensionl nlysis migt elp students see tt te eigt is 3.1 m m, or 3.1m. To elp students understnd squre units, [Ask] Wt is te re of rectngle wit dimensions 3 meters y 4 meters? [1 squre meters (m )] Wt is 1 meters squred? [It represents te re of squre 1 m on side, wic is 144 m.] 48 cm 1 m 11 m 9 ft 7 ft 1 ft 1 m.3 m 5 m [Ask] How would you ceck te nswer to Emple B? [You could multiply eigt times se to see tt you get 7.13 m.] Assessing Progress You cn ssess students understnding of rectngles, prllelogrms, nd tringles, s well s teir comfort wit te conservtion of re s figure is roken into pieces tt re ten rerrnged into noter spe. Sring Ides (continued) opposite sides of te prllelogrm is congruent to pir of opposite sides of te rectngle, te eigts nd res re te sme, ut te ngles nd nonse sides of te prllelogrm re not congruent to teir corresponding prts on te rectngle.] [Alert] Students cn mesure te side lengts to find te re of rectngle, ut te side lengts of prllelogrm don t necessrily elp tem find its re; tey must determine te eigt. [Ask] Is it ever te cse tt side lengt equls te eigt? [yes, for rectngle] If you rotte te prllelogrm, ow does its re cnge? Don t ccept too quickly te nswer It doesn t nd move on. Some students my e unsure. [Ask] Wt do you tink of clim tt te re of ny prllelogrm is zero, ecuse te prllelogrm consists only of four line segments, ec of wic s re zero? Altoug we usully refer to te re of geometric figure, we men te re of te region inside te figure. [Lnguge] In Englis, te word re often mens region. Suggest tt students use te word region in tis contet so tt tey won t e sying tings like te re of te re. Pose te reverse prolem: If we know te re, cn we find te se nd eigt? Or If we know te re nd se, cn we find te eigt? Discussion of tis question cn led to Emple B. [Ask] Do ll rectngles wit te sme perimeter ve te sme re? Students migt tink tey do, yet not sy so. Drw pictures or use loop of string to form rectngles tt get longer nd nrrower wile mintining te sme perimeter. At some point, most students will decide tt te re does cnge. A few my need to do clcultions to e convinced. LESSON 8.1 Ares of Rectngles nd Prllelogrms 45

8 Closing te Lesson Restte te mjor points of te lesson: Te re of figure is mesure of te region inside te figure. If te figure is roken into prts, te re of te originl figure is te sum of te res of tose prts. Are is mesured in squre units nd is te numer of unit squres tt cn e fit into te figure, perps fter it s een roken into pieces nd ressemled. Te res of rectngle nd prllelogrm cn e clculted y multiplying te lengt of te se y te eigt, were ny side cn e considered te se nd te eigt is te lengt of n ltitude to tt se. 10. Sketc nd lel two different rectngles, ec wit re 48 cm. In Eercises 11 13, find te sded re nd eplin your metod squre squre 13. units units 14. Sketc nd lel two different prllelogrms, ec wit re 64 cm. 15. Drw nd lel figure wit re 64 cm nd perimeter 64 cm. 16. Te poto sows Jpnese police kon. An rc forms prt of te roof nd one wll. Te rc is mde from rectngulr pnels tt ec mesure 1 m y 0.7 m. Te rc is 11 pnels ig nd 3 pnels wide. Wt s te totl re of te rc? 3.1 m Culturl 7 1 squre units BUILDING UNDERSTANDING Kon is Jpnese for mini-sttion, smll police sttion. Tese sttions re locted in severl prts of city, nd officers wo work in tem know te surrounding neigoroods nd people well. Te presence of kons in Jpn elps reduce crime nd provides communities wit sense of security. Te eercises provide opportunities to prctice pplying te re concepts. Hve grp pper ville nd encourge students to mke ccurte drwings s prt of teir solutions. ASSIGNING HOMEWORK Essentil 1 15, 17, 0 3 Performnce ssessment 19, 5 Portfolio 18 Group 16, 19 4 Review 7 9 Alger review 3 5, 8, 19 MATERIALS Eercise 7 (T), optionl 17. Wt is te totl re of te four wlls of rectngulr room 4 meters long y 5.5 meters wide y 3 meters ig? Ignore ll doors nd windows. (4)(3) (5.5)(3) 57 m 18. Appliction Ernesto plns to uild pen for is pet igun. Wt is te re of te lrgest rectngulr pen tt e cn mke wit 100 meters of fencing? For constnt perimeter, re is mimized y squre. 100 m 4 5 m per side; A 65 m. 19. Te ig event t George Wsington Hig Scool s My Festivl ec yer is te Cow Drop Contest. A frmer rings is well-fed ovine to wnder te footll field until well, you get te picture. Before te contest, te footll field, wic mesures 53 yrds wide y 100 yrds long, is divided into squre yrds. Scool clus nd clsses my purcse squre yrds. If one of teir squres is were te first dropping lnds, tey win pizz prty. If te mt clu purcses 10 squres, wt is te proility tt te clu will win? Appliction Sr is tiling wll in er troom. It is rectngulr nd mesures 4 feet y 7 feet. Te tiles re squre nd mesure 6 inces on ec side. How mny tiles does Sr need? 11 Helping wit te Eercises Eercise 5 [Alert] Students my confuse perimeter wit re. Eercise 7 [Alert] Students migt multiply te eigt y te noncorresponding se. Eercise 9 Remind students tt te ltitude is not lwys in te interior of figure. Eercises 10, 14, smple nswer: 4 cm 1 cm 48 cm Encourge vriety of nswers. 8 cm 6 cm 48 cm Eercises Student eplntions of teir metods my include counting squres nd frctions of squres. 14. smple nswers: 64 cm 16 cm 4 cm 8 cm 64 cm 8 cm Eercise 15 Tere re mny solutions, including very skinny prllelogrm wit se nd eigt 8 nd oter sides wit lengt possile nswer: 16 cm 16 cm 16 cm 4 cm 16 cm 46 CHAPTER 8 Are

9 For Eercises 1 3, you my coose to use grid pges. 1. Find te re of qudrilterl ABCD wit vertices A(0, 0), B(6, 0), C(14, 16), nd D(8, 16). 96 squre units. Find te re of qudrilterl EFGH wit vertices E(0, 0), F(6, 4), G(8, 0), nd H(6, 4). 3 squre units 3. Find te re of te trpezoid t rigt. 500 cm 5 cm 4. Mini-Investigtion Suppose you mesure te sides of rectngle to clculte its re. Using your ruler, you find tt te se of te rectngle is etween 15.6 cm nd 15.7 cm, nd te eigt is etween 1.3 cm nd 1.4 cm.. Find te smllest possile re nd te lrgest possile re. smllest: cm ; lrgest: cm. Wt do you tink is resonle vlue for te ctul re of te rectngle? Answers will vry. Smple nswer: out 193 cm c. If you use cm for te se nd 1.35 cm for te eigt, te clculted re. is cm. From your nswers to prt, eplin wy te lst four digits in tis clcultion re not considered significnt. Answers will vry. Te smllest nd lrgest re vlues differ t te ones plce, so te digits fter te deciml point re insignificnt compred to te effect of te limit of precision in te mesurements. Art 1 cm Te design t rigt is quilt design clled te Oio Str. Trditionl quilt lock designs rnge from simple nine-ptc, sed on 9 squres, to more complicted designs like Jco s Ldder or Underground Rilrod, wic re sed on 16, 5, or even 36 squres. Quiltmkers need to clculte te totl re of ec different type of mteril efore tey mke complete quilt. 1 cm 0 cm 5. Appliction Te Oio Str is 16-squre quilt design. Ec lock mesures 1 inces y 1 inces. One lock is sown ove. Assume you will need n dditionl 0% of ec fric to llow for sems nd errors.. Clculte te sum of te res of ll te red ptces, te sum of te res of ll te lue ptces, nd te re of te yellow ptc in single lock.. How mny Oio Str locks will you need to cover n re tt mesures 7 inces y 84 inces, te top surfce re of king-size mttress? 4 c. How muc fric of ec color will you need? How muc fric will you need for 15-inc order to etend eyond te edges of te top surfce of te mttress? 6 6. A rigt tringle wit sides mesuring 6 cm, 8 cm, nd 10 cm 10 s squre constructed on ec of its tree sides, s sown. 8 Compre te re of te squre on te longest side to te sum of te res of te two squres on te two sorter sides. 100; Te re of te squre on te longer side is te sme s te sum of te res on te oter two legs. 5c. Aout 1814 in of red fric, out 369 in of lue fric, nd out 1814 in of yellow fric. Te order requires 5580 in (if it does not need te etr 0%). Eercise 7 Encourge students to strt smll, wit wt tey cn conclude from te given informtion. After tey egin, teir iggest cllenge my e to find vlues for nd. As needed, focus teir ttention on nd te inscried ngle DBC. Once tey see tt te ltter is rigt ngle, tey cn proceed in severl wys. Te vlue of is te difference etween rigt ngle (y te Tngent Segments Conjecture) nd te ngle t B of smll tringle, s is te given ngle mesure, 5. Or tey cn first find c nd ten find te ngle mesure of te smll tringle y te Isosceles Tringle nd Tringle Sum Conjectures. Te Solutions Mnul includes complete eplntion of ow to find ec ngle. Eercise 18 Encourge students to try vrious ses nd eigts nd to look for pttern in tose dimensions tt give te lrgest re nd tose tt give te smllest, or to grp se versus re. [Link] If students grp se versus re, tey migt notice tt te function ppers qudrtic. Eercise 19 Tis eercise connects re wit proility. Geometric proility is clculted y dividing te re of te selected region y te re of te entire region. Eercise 1 Students migt use te formul for te re of prllelogrm, or tey migt count unit squres. Eercise Are formuls for tringles nd kites re developed in Lesson 8.. At tis point students migt count unit squres, or tey migt divide te kite long its digonls into four rigt tringles. Te ltter metod cn e generlized into te formul for te re of kite. Eercise 4 Tis mini-investigtion ddresses te reltionsip etween te precision of mesurements nd te numer of significnt digits in clcultions using tose mesurements. Unless oterwise indicted, te mesures given on figures in te ook re ssumed to e ect. However, wen students use ruler to mesure lengts, nd ten use tose lengts in clcultions, you migt prompt tem to consider te pproprite numer of significnt digits to use in teir results. 5. In one Oio Str lock, te sum of te red ptces is 36 in, te sum of te lue ptces is 7 in,nd te yellow ptc is 36 in. LESSON 8.1 Ares of Rectngles nd Prllelogrms 47

10 Eercise 7 See tecing note t te ottom of pge 47. Eercise 8 If students re ving difficulty locting te sides of te tringle tt ren t perpendiculr to te ltitude, sk wt ngles tey mke wit te ltitude [30 ] nd ow to find tose ngles [isect 60 ngle]. 8. smple construction: Cpter Review D d 7. Developing Proof Copy te figure t rigt. Find te 94 lettered ngle mesures nd rc mesures. AB nd AC re tngents. CD is dimeter. Eplin ow you g determined te mesures nd. E 76, 5, c 104, d 5, e 76, f 47, g 90, 43, k 104, m 86. Eplntions will vry. 8. Given AM s te lengt of te ltitude of n equilterl tringle, construct te tringle. A M B 5 e c m F f C k A A M Sketc wt te figure t rigt looks like wen viewed from ec of tese directions.. Aove te figure, looking strigt down. In front of te figure, tt is, looking strigt t te red-sded side c. Te side, looking t te lue-sded side EXTENSIONS A. Students migt enjoy creting geoord cllenges for ec oter or for oter groups. One student (or group) cretes n irregulr spe on geoord nd cllenges noter student (or group) to find its re. B. Pose tis prolem: Find rectngles wit integer side lengts in wic te numer of squre units in te re equls te numer of units in te perimeter. [For students wo find te two nswers, 4 y 4 nd 3 y 6, [Ask] Wy re tose te only two rectngles? ] C. Estimte te re of Tennessee, Ut, or Wyoming. [pproimte res: Tennessee: 4,000 mi ; Ut: 85,000 mi ; Wyoming: 98,000 mi ] Project If your students need more guidnce wit Ftom, you migt use te Are nd Perimeter ctivity from Tecing Mtemtics wit Ftom. See pge 841 for nswers to Eercises 9,, nd c nd grps for te project. RANDOM RECTANGLES Wt does typicl rectngle look like? A rndomly generted rectngle could e long nd nrrow, or squre-like. It could ve lrge perimeter, ut smll re. Or it could ve lrge re, ut smll perimeter. In tis project you will rndomly generte rectngles nd study teir crcteristics using sctter plots nd istogrms. Your project sould include A description of ow you creted your rndom rectngles, including ny constrints you used. A sctter plot of se versus eigt, perimeter istogrm, n re istogrm, nd sctter plot of re versus perimeter. Any oter studies or grps you tink migt e interesting. Your predictions out te dt efore you mde ec grp. An eplntion of wy ec grp looks te wy it does. Supporting te You cn lso use grping clcultor to do tis project. Students my need to e reminded tt istogrm is similr to r grp wit te rs djcent; ec r represents rnge of vlues. [Ask] How mny rectngles re enoug? Discuss smple size. You cn use Ftom to generte rndom se nd eigt vlues from 0 to 10. Ten you cn sort tem y vrious crcteristics nd mke wide rnge of interesting grps. OUTCOMES Student provides cler descriptions, constrints, eplntions, nd predictions. Student includes grps similr to tose in te nswer section on pge 841. Oter grps nd reltionsips re presented. 48 CHAPTER 8 Are

11 LESSON 8. Wen you dd to te trut, you sutrct from it. THE TALMUD You will need evy pper or crdord You will need evy pper or crdord Step 1 Step Step 3 Step 1 Step Ares of Tringles, Trpezoids, nd Kites In Lesson 8.1, you lerned te re formul for rectngles, nd you used it to discover n re formul for prllelogrms. In tis lesson you will use tose formuls to discover or demonstrte te formuls for te res of tringles, trpezoids, nd kites. Investigtion 1 Are Formul for Tringles s d Cut out tringle nd lel its prts s sown. Mke nd lel copy. Arrnge te tringles to form figure for wic you lredy ve n re formul. Clculte te re of te figure. Wt is te re of one of te tringles? Mke conjecture. Write rief description in your noteook of ow you rrived t te formul. Include n illustrtion. C-76 Tringle Are Conjecture A 1 Te re of tringle is given y te formul?,were A is te re, is te lengt of te se, nd is te eigt of te tringle. Investigtion Are Formul for Trpezoids 1 s Construct ny trpezoid nd n ltitude perpendiculr to its ses. Lel te trpezoid s sown. Cut out te trpezoid. Mke nd lel copy. LESSON OBJECTIVES Derive formuls for te res of tringles, trpezoids, nd kites Apply re formuls to solve prolems Develop coopertive evior LESSON 8. PLANNING LESSON OUTLINE One dy: 5 min Investigtion 15 min Sring nd Closing 5min MATERIALS Eercises evy pper or crdord scissors Sketcpd demonstrtion Ares of Tringles nd Trpezoids, optionl Sketcpd ctivity Ares of Tringles, Trpezoids, nd Kites, optionl TEACHING If you don t use te one-step investigtion (pge 430), you migt jigsw te investigtions. [ELL] Add te terms trpezoid, tringle, nd kite to te cpter voculry disply. Guiding Investigtion 1 You migt do Investigtion 1 s follow-long ctivity. Step 1 Encourge students in ec group to mke different kinds of tringles. Step 3 [Alert] Students my ve difficulty going ckwrd from te re of two tringles to te re of one. Some students my know tt te re of rigt tringle is lf te re of rectngle. Ask tem to eplore weter te formul generlizes to ll tringles. Guiding Investigtion Step 1 Students sould mke vriety of kinds of trpezoids, including rigt nd isosceles trpezoids. LESSON 8. Ares of Tringles, Trpezoids, nd Kites 49

12 Step 3 Students my e confused y te numer of vriles needed to write te formul. If necessry, [Ask] Wt lengts need vrile nmes? Wt re some good nmes for tose lengts? [Looking t te Trpezoid Are Conjecture will elp.] One Step To comine te investigtions, pose tis prolem: Find formuls for te re of tringles, trpezoids, nd kites. Mny students my lredy know formul for te re of tringle; encourge tem to derive it from te re formuls for oter figures y copying, cutting, nd ssemling tringles. Encourge vriety of pproces. For emple, some students my derive te formul for te re of trpezoid y dividing it into pieces, oters y putting two copies end to end, nd oters y mking rectngle oftesmerewoselengtis te midsegment. Guiding Investigtion 3 If students re stuck, [Ask] Is it elpful to use some properties of kites tt you know? [Properties include tt one digonl divides te kite into two congruent tringles, te oter digonl divides te kite into two isosceles tringles, nd te digonls of kite re perpendiculr to ec oter nd tus divide te kite into four rigt tringles.] Encourge cretivity. Students my wis to turn over two of te four tringulr pieces. Oters my consider te kite s rectngle wit tringulr pieces removed. Oters my tink of te kite s two congruent tringles. As needed, sk weter students re using te digonls or prts of te digonls. Remind students to define te formul s vriles witin te conjecture, s in ll te re conjectures. Step 3 Step 3 prllelogrm wit se 1 ; 1 flourisk.com EXERCISES Trpezoid Are Conjecture Te re of trpezoid is given y te formul?,were A is te re, 1 nd re te lengts of te two ses, nd is te eigt of te trpezoid. Investigtion 3 Are Formul for Kites Cn you rerrnge kite into spes for wic you lredy ve te re formul? Do you recll some of te properties of kite? Crete nd crry out your own investigtion to discover formul for te re of kite. Discuss your results wit your group. Stte conjecture. In Eercises 1 1, use your new re conjectures to solve for te unknown mesures. 1. A? 0 cm. A? 49.5 m 3. A? 300 squre units 5 cm 6 cm 15 9 m 4. A? 60 cm 5. A 39 cm 6. A 31.5 ft? 6 cm? 9 ft 8 cm 6 cm 14 cm 8 cm 6 cm SHARING IDEAS As students sre ides out te Tringle Are Conjecture, [Ask] How do you know wt side is te se of te tringle? Try joining different sides to see ow te resulting prllelogrms differ. Students my ve difficulty seeing tt tese tree prllelogrms ve te sme re. Suggest tt tey look t ec se-ltitude pir of one prllelogrm nd see ow it s se-ltitude pir of noter one of te prllelogrms. Arrnge te two trpezoids to form figure for wic you lredy ve n re formul. Wt type of polygon is tis? Wt is its re? Wt is te re of one trpezoid? Stte conjecture. [ To view interctive versions of tese investigtions, see te Dynmic Geometry Eplortion Ares of Tringles, Trpezoids, nd Kites t flourisk.com. ] 11 m 13 cm A 1 1 C-78 Kite Are Conjecture A 1 d 1 d,were d 1 nd d re Te re of kite is given y te formul?. te lengts of te digonls ft ft 0 C-77 For te Trpezoid Are Conjecture, [Ask] Is prllelogrm formed for ny trpezoid? How do you know? [Possile pproces include sowing opposite sides or opposite ngles re congruent.] Does it mke difference wic se is leled 1? [No; teir sum is te sme in te prllelogrm.] Does 1 1 sound fmilir to you? [It s te verge of te lengts of te ses; it s te lengt of te trpezoid s midsegment.] 430 CHAPTER 8 Are

13 7. A 40 ft 8. A 50 cm 9. A 180 m LE? 30 ft? 5 cm? E 16 m 7 cm 9 m 17 ft U 6 cm 4 m 5 ft 8 ft 17 ft 13 cm 0 ft 10. A 94 cm 11. A 04 cm 1.? P? 168 cm P 6 cm y?? 1 cm 51 cm 13. Sketc nd lel two different tringles, ec wit re 54 cm. 14. Sketc nd lel two different trpezoids, ec wit re 56 cm. 15. Sketc nd lel two different kites, ec wit re 109 cm. 16. Sketc nd lel tringle nd trpezoid wit equl res nd equl eigts. How does te se of te tringle compre wit te two ses of te trpezoid? 17. P is rndom point on side AY of 18. One plying crd is plced over noter, rectngle ARTY. Te sded re is wt s sown. Is te top crd covering lf, frction of te re of te rectngle? Wy? less tn lf, or more tn lf of te Y T 1 ottom crd? Eplin. more tn lf, ecuse (To see wy, drw te top crd completely ltitude PQ.) covers one corner of te ottom crd P B A 5 ft 4 cm 40 cm L 19. Appliction Edurdo s designed tis kite for contest. He plns to cut te kite from seet of Mylr plstic nd use ls wood for te digonls. He will connect ll te vertices wit string, nd fold nd glue flps over te string.. How muc ls wood nd Mylr will e need?. Mylr is sold in rolls 36 inces wide. Wt lengt of Mylr does Edurdo need for tis kite? R 15 cm Assessing Progress As you oserve students working nd presenting, ssess teir understnding of te res of rectngles nd prllelogrms. You cn lso ceck teir understnding of re s te numer of squres tt will cover te spe nd teir comfort wit te conservtion of re. 10 cm 1 in. 13 cm 0 in. 5 in. 1 in. 15 in. A 5 ft Closing te Lesson 6 ft 39 in. 3.6 ft 10.8 ft B 36 in. 15 ft 35 in. Summrize te mjor points of tis lesson: Every tringle is lf prllelogrm wit te sme se nd eigt, so te re of tringle is 1. Every trpezoid is lf prllelogrm wose se is te sum of te ses of te trpezoid nd wose eigt is te sme, so te re of trpezoid is 1 1 ; tis is te sme s te lengt of te midsegment times te eigt. Every kite is lf rectngle wose se nd eigt re te kite s digonls, so te re of kite is 1 d 1 d. y 9 ft C Sring Ides (continued) [Ask] Wt ppens to te re of trpezoid if one of te ses srinks to lengt zero? [Te trpezoid ecomes tringle wit te sme eigt, te midsegment ecomes midsegment of te tringle, nd te re ecomes lf te remining se times te eigt.] BUILDING UNDERSTANDING Students my continue to rek figures into fmilir spes rter tn just using te formul in teir conjecture. As needed, you migt work severl eercises s demonstrtions. ASSIGNING HOMEWORK Essentil 1 15 Performnce ssessment 19, 0 Portfolio 14 Journl 16 18, 1, Group 18 Review 3 31 Alger review 5 1, 19 MATERIALS Eercise 30 (T), optionl Helping wit te Eercises Eercises 1 1 [Alert] Mny of te pictures include etrneous informtion, so students my use irrelevnt numers. Eercise 7 Students my use eiter te Kite Are Conjecture or te Tringle Are Conjecture. Eercise 1 Tis is cllenging eercise. Students my e confused ytedsedlinesndnotrelize tt tey re ltitudes. Wonder loud weter tere re different wys to epress te re of te tringle. 1 (6)(9), 1 (5)y, 1 (15) See pges for nswers to Eercises 13 16, nd 19. LESSON 8. Ares of Tringles, Trpezoids, nd Kites 431

14 Eercise 13 [Alert] Students my look for wole numers wose product is 54 rter tn numers lf of wose product is 54. Eercise 18 If students re ving difficulty, sk weter tis picture reltes to tt of Eercise 17. Te rounded edges cn e ignored. Eercise 1 Students my e confused ecuse te trpezoid s ses ren t orizontl. To preview te Pytgoren Teorem, [Ask] Crete n eqution of te two re formuls. After multiplying nd comining like terms, wt is te eqution? 1 ( )( ) 1 c 1 1 c c c Tis proof is ttriuted to U.S. President Jmes Grfield. 1. Te isosceles tringle is rigt tringle ecuse te ngles on eiter side of te rigt ngle re complementry. If you use te trpezoid re formul, te re of te trpezoid is 1 ( )( ). If you dd te res of te tree tringles, te re of te trpezoid is 1 c. Eercise Tis pproc my repet wt students ve lredy done. If not, [Ask] How does te nswer compre wit te formul found in Investigtion? Eercise 30 Tis puzzle is similr to Lesson 8.1, Eercise 7. Students my ve troule finding d, e, nd g. Encourge tem to look for congruent tringles (to sow d e, so tey cn use te fct tt d e mbc ) Appliction Te roof on Crystl s ouse is formed y two congruent trpezoids nd two congruent isosceles tringles, s sown. Se wnts to put new wood singles on er roof. Ec single will cover 0.5 squre foot of re. (Te singles re 1 foot y 1 foot, ut tey overlp y 0.75 squre foot.) How mny singles sould Crystl uy? 3600 singles to cover n re of 900 ft 1. A trpezoid s een creted y comining two congruent rigt tringles nd n isosceles tringle, s sown. Is te isosceles tringle rigt tringle? How do you know? Find te re of te trpezoid two wys: first y using te trpezoid re formul, nd ten y finding te sum of te res of te tree tringles. Review 3. A? 4. A? 5. A 64 m 6. P 5 cm P? 70 m A? 144 cm 7. A? 88 ft A? P? 144 ft P? 18 EXTENSIONS. Divide trpezoid into two tringles. Use lger to derive te formul for te re A 1 of te trpezoid y epressing te re of ec tringle lgericlly nd finding teir lgeric sum. Given: trpezoid ABCD wit eigt. re of ABD 1 1 ;re ofbcd 1 ;re oftrpezoid sum of res of D two tringles squre units 7 squre units cm 40 cm 8 cm 9. Identify te point of concurrency from te construction mrks... c. A. Develop te formul for te res of tringle nd trpezoid s (midsegment) (eigt). B. Find more metods for rriving t te formul for te re of trpezoid. Consider wys of dividing te trpezoid into pieces (tringles, rectngles, prllelogrms) for wic you lredy know te re formul ft 4 m 15 ft c 10 cm 9 cm 8 cm 0 ft 15 ft 40 cm c 30 ft 10 cm possile nswers (ll use trpezoid ABCD s sown): A D cm 0 cm incenter ortocenter centroid C re re(abc) re(cda) B B C 11 cm 43 CHAPTER 8 Are

15 30. Developing Proof Trce te figure elow Give te verte rrngement for tis -uniform Find te lettered ngle mesures nd rc tesselltion. 3.6 / mesures. AB nd AC re tngents. CD is dimeter. Eplin ow you determined te mesures d nd n. B g A D m c d f n e p C 56 E , 68, c 68, d 56, e 56, f 90, g 34, 56, m 56, n 90, p 34. Possile eplntion: Let O e te center of te circle. mbc 11 y te Inscried Angle Conjecture, nd d e mbc y te Centrl Angle Conjecture. OBA is congruent to OCA y SSS, so d e 56. DEC is semicircle, so mde 68. By te Inscried Angle Conjecture, p 34. Using OEC nd te Tringle Sum Conjecture, n 90. Etension B (continued) Construct point E on AB so tt ED is prllel to BC. A D re Construct DE AB nd CF AB, wit points E nd F on AB. Let c AE. D E 1 C c A E 1 F C B B MAXIMIZING AREA A frmer wnts to fence in rectngulr pen using te wll of rn for one side of te pen nd te 10 meters of fencing for te remining tree sides. Wt dimensions will give er te mimum re for te pen? You cn use te trce feture on your clcultor to find te vlue of tt gives te mimum re. Use your grping clcultor to investigte tis prolem nd find te est rrngement. re 1 c 1 1 c 1 1 E' A D 1 Strt s you did in te previous metod, ten duplicte AED s DEA nd FBC s FCB, forming rectngle ABFE. C F' B re 1 1 c 1 1 c 1 1 X = 3.5 Y = 10.5 Your project sould include An epression for te tird side lengt, in terms of te vrile in te digrm. An eqution nd grp for te re of te pen. Pen C. Estimte te re of Cliforni, Tes, or Nevd. [pproimte res: Cliforni: 164,000 mi ; Tes: 69,000 mi ; Nevd: 111,000 mi ] Te dimensions of te est rectngulr spe for te frmer s pen. Brn wll Supporting te If tey wis, students migt gter some dt points nd use qudrtic regression rter tn finding nd grping function epression. OUTCOMES Te epression nd eqution re equivlent to A (10 ). Te grp sows te mimum re t.5. Te project includes n eplntion of wy te mimum re is 1.5 m.one reson, not using te grp, is tt if tere were twice s muc fencing ville nd no rn, ten te region of mimum re would e squre (s discovered in Lesson 8.1, Eercise 18); te rn wll cn e considered line of symmetry, cutting te mount of fencinginlfutmkingtespe lf squre, wit widt 1 4 te totl mount of fencing. LESSON 8. Ares of Tringles, Trpezoids, nd Kites 433

16 LESSON 8.3 PLANNING LESSON OUTLINE One dy: 30 min Investigtion 10 min Sring nd Closing 5min MATERIALS rulers clcultors Eercises Finding Ares (T) TEACHING Ares of vriety of spes cn e found if you know ow to find te res of te stndrd spes of te previous lessons. LESSON 8.3 Optimists look for solutions, pessimists look for ecuses. SUE SWENSON You will need figures A H centimeter rulers or meterstick Are Prolems By now, you know formuls for finding te res of rectngles, prllelogrms, tringles, trpezoids, nd kites. Now let s see if you cn use tese re formuls to pproimte te res of irregulrly sped figures. A D B C Investigtion Solving Prolems wit Are Formuls Find te re of ec geometric figure your tecer provides. Before you egin to mesure, discuss wit your group te est strtegy for ec step. Discuss wt units you sould use. Different group memers migt get different results. However, your results sould e close. You my verge your results to rrive t one group nswer. For ec figure, write sentence or two eplining ow you mesured te re nd ow ccurte you tink it is. E F H G Guiding te Investigtion One Step Pose tis prolem, tken from Discovering Alger: You re surveying four-sided piece of lnd. From your Glol Positioning System (GPS), you ve found tt one corner of te lnd is t tree 638 feet due nort of te corner tt s net to te lrge rock. A tird corner is t post 138 feet sout nd 550 feet est of tt tree. Te fourt corner is t te edge of pond. Te rock is 400 feet west nd 100 feet nort of te corner y te pond. Wt s te re of te plot of lnd? As you circulte, encourge grping nd viewing te spe in lots of different wys: s te union of fmilir spes (including smll squres), s te result of tking wy spes from rectngle, nd s pproimtely rigt trpezoid. Now tt you ve prcticed mesuring nd clculting re, you re redy to try some ppliction prolems. Mny everydy projects require you to find te res of flt surfces on tree-dimensionl ojects. You ll lern more out surfce re in Lesson 8.7. Creer Professionl ousepinters ve unique comintion of skills: For lrge-scle jos, tey egin y mesuring te surfces tt tey will pint nd use mesurements to estimte te quntity of mterils tey will need. Tey remove old coting, clen te surfce, pply seler, mi color, pply pint, nd dd finises. Pinters ecome eperienced nd specilize teir crft troug te on-te-jo trining tey receive during teir pprenticesips. In preprtion for te investigtion, use te two Finding Ares trnsprencies to project te imges onto lrge seets of utcer pper nd mke lrge copies tt cn e reused ec yer. An lterntive is to use ctul nonrectngulr regions in your clssroom if tey eist nd re esily mesured. Te tsk of ec group is to find te re of te figures ssigned. Encourge groups to tink out wt is te est strtegy. Sred tsks nd tem effort re importnt. As students work, encourge vriety of pproces. For emple, some students my decide to sudivide Figure G into congruent tringles, wile oters will divide it into qudrilterls nd tringles. Similrly, Figure H prticulrly lends itself to vriety of pproces. Offer dvice only if student ecomes frustrted to te point of giving up; one gol is to ve students feel te eilrtion of meeting cllenges. Groups or students wo wnt to etend te investigtion cn lso find te perimeter of ec spe. 434 CHAPTER 8 Are

17 EXERCISES 1. Appliction Tmmy is estimting ow muc se sould crge for pinting 148 rooms in new motel wit one cot of se pint nd one cot of finising pint. Te four wlls nd te ceiling of ec room must e pinted. Ec room mesures 14 ft y 16 ft y 10 ft ig.. Clculte te totl re of ll te surfces to e pinted wit ec cot. Ignore doors nd windows. 11,95 ft. One gllon of se pint covers 500 squre feet. One gllon of finising pint covers 50 squre feet. How mny gllons of ec will Tmmy need for te jo? 44 gl of se pint nd 488 gl of finising pint. Appliction Rsd wnts to wllpper te four wlls of is edroom. Te room is rectngulr nd mesures 11 feet y 13 feet. Te ceiling is 10 feet ig. A roll of wllpper t te store is.5 feet wide nd 50 feet long. How mny rolls sould e uy? (Wllpper is ung from ceiling to floor. Ignore doors nd windows.) He sould uy t lest four rolls of wllpper. (Te re of ec roll is 15 ft. Te totl surfce re to e ppered is 480 ft.) If pper cut off t te corners is wsted, e ll need 5 rolls. 3. Appliction It tkes 65,000 solr cells, ec 1.5 in. y.75 in., to power te Helios Prototype, sown elow. How muc surfce re, in squre feet, must e covered wit te cells? Te cells on Helios re 18% efficient. Suppose tey were only 1% efficient, like solr cells used in omes. How muc more surfce re would need to e covered to deliver te sme mount of power? 155 ft ; 776 ft more surfce re Tecnology In August 001, te Helios Prototype, remotely controlled, nonpolluting solr-powered ircrft, reced 96,500 feet record for nonrocket ircrft. Soon, te Helios will likely sustin fligt long enoug to enle weter monitoring nd oter stellite functions. For news nd updtes, go to flourisk.com. In te eercises you will lern ow to use re in uying rolls of wllpper, gllons of pint, undles of singles, squre yrds of crpet, nd squre feet of tile. Keep in mind tt you cn t uy gllons of pint! You must uy 13 gllons. If your clcultions tell you tt you need 5.5 undles of singles, you ve to uy 6 undles. In tis type of rounding, you must lwys round upwrd. LESSON OBJECTIVES Prctice mesuring Prctice estimtion Solve re ppliction prolems using vrious prolemsolving strtegies Develop reding compreension nd coopertive evior SHARING IDEAS Tis is good cnce to ve presenttions from students wo ren t lwys successful nd ve found vlues for te res of one or more of te first five figures. Figures F nd G motivte Lessons 8.4 nd 8.5. Look t inscried nd circumscried figures in Figure H. Answers for ll figures my vry. Some students, despite ving centimeter rulers ville, migt use oter mesuring units. Or differences in precision migt led to different numers. [Ask] Does tis men tt figure cn ve different res? [For ny mesurement unit, ec figure s unique re. (An re function ssigns ec spe single nonnegtive numer.) How close te re you find is to tt unique re depends on te precision of your mesurement.] You migt monitor te use of significnt digits. For emple, if te se nd eigt of tringle re mesured to e 13.6 cm nd 18 cm, respectively, ten te re is ccurte to two significnt digits, 10 cm,rter tn 1.4 cm or 1 cm.ifot mesurements were tken to te sme level of ccurcy, 18.0 cm sould ve een stted, nd te nswer would e 1 cm. [Ask] Does n interior decortor need to use more or less precision wen estimting ow muc crpet is needed for ome tn crpet lyer needs to use wen mesuring ow to cut crpet for single room? [Te interior decortor cn e less precise.] How cn you get more precise results? [mesuring wit more precise tools; reding te mesurement to smller unit] Assessing Progress Assess ow well students recognize nd find te res of rectngles, prllelogrms, tringles, trpezoids, nd kites. Also ceck up on temwork skills. LESSON 8.3 Are Prolems 435

18 Closing te Lesson Being le to find te res of some sic spes llows you to find, or t lest pproimte, te res of wide vriety of spes. To elp find res of figures suc s F nd G in te investigtion, future lessons will investigte finding te res of oter sic spes. BUILDING UNDERSTANDING Introduce te eercise set y tlking out situtions in wic it s pproprite to round ll numers up. If 160 students re trveling in uses tt crry 70, ten you need 3 uses, not 7. Toug te set contins only 1 prolems, doing tem ll will tke lot of time. You migt ve different groups work on different prolems nd prepre to present teir ides to te clss. [ELL] You migt ve Englis lerners work in pirs on te eercises. Students cn trde off etween reding te prolem loud to teir prtner nd eplining wt tey tink te prolem mens. Students cn drw digrm togeter, nd ten solve te prolem independently nd compre results. ASSIGNING HOMEWORK Essentil 1 9 Portfolio 4 Group 1 9 Review 10 1 Alger review 1 4, 6, 7, 10 Helping wit te Eercises Eercise 1 As needed, encourge students to drw top view of te room. Eercise 3 Students migt reson tt if te cells re 3 s efficient ten tere will e 3 s mny. 4. Appliction Hrold works t stte prk. He needs to sel te redwood deck t te informtion center to protect te wood. He mesures te deck nd finds tt it is kite wit digonls 40 feet nd 70 feet. Ec gllon of selnt covers 400 squre feet, nd te selnt needs to e pplied every si monts. How mny gllon continers sould e uy to protect te deck for te net tree yers? 1 5. Appliction A lndscpe rcitect is designing tree trpezoidl flowereds to wrp round tree sides of egonl flgstone ptio, s sown. Wt is te re of te entire flowered? Te lndscpe rcitect s fee is $100 plus $5 per squre foot. Wt will te flowered cost? 336 ft ; $1780 Creer Lndscpe rcitects ve keen eye for nturl euty. Tey study te grde nd direction of lnd slopes, stility of te soil, dringe ptterns, nd eisting structures nd vegettion. Tey look t te vrious socil, economic, nd rtistic concerns of te client. Tey lso use science nd engineering to pln environments tt rmonize lnd fetures wit structures, reducing te impct of urn development upon nture. For Eercises 6 nd 7, refer to te floor pln t rigt. 6. Appliction Dreen s fmily is redy to ve wll-to-wll crpeting instlled. Te crpeting tey cose costs $14 per squre yrd, te pdding $3 per squre yrd, nd te instlltion $3 per squre yrd. Wt will it cost tem to crpet te tree edrooms nd te llwy sown? $ Appliction Dreen s fmily now wnts to instll 1-foot-squre terr-cott tiles in te entrywy nd kitcen, nd 4-inc-squre lue tiles on ec troom floor. Te terr-cott tiles cost $5 ec, nd te troom tiles cost 45 ec. How mny of ec kind will tey need? Wt will ll te tiles cost? 0 terr cott tiles, 1107 lue tiles; $ Eercise 4 Tis is good prolem to model for te clss; it migt generte some interesting disgreements. Some students my miss te condition for te net tree yers. Most students my ssume tt selnt left over fter one ppliction cn e used si monts lter. Tis possiility sould e discussed nd greed upon s clss. Eercise 6 Students my e confused out te widts of llwys. Wonder loud weter tose widts cn e found from given dimensions. 7 ft 40 ft 70 ft Bedroom 10 ft 9 ft Bedroom 10 ft 8 ft Entrywy Kitcen 10 ft 18 ft 1 ft 0 ft Bt 10 ft 6 ft Bt 7 ft 9 ft Hllwy Mster edroom 13 ft 8 ft Living room 13 ft 13 ft Dining room 13 ft 9 ft [Alert] Students my ve difficulty cnging squre feet to squre yrds. Suggest tt tey drw squre yrd nd sudivide it into squre feet. Eercise 7 If students re not sure ow to convert etween squre feet nd squre inces, gin ve tem drw picture. 436 CHAPTER 8 Are

19 Find te re of te sded region. 9. CE, BD, nd AF re ltitudes. Find AB nd BD. Assume ll te squres re congruent. AB 16.5 cm, BD 15.3 cm 7 cm Review 1 cm 1 cm AC 11.0 cm 10. A first-century Greek mtemticin nmed Hero is credited wit te following formul for te re of tringle: A s(s )(s )(s ), c were A is te re of te tringle,,, nd c re te lengts of te tree sides of te tringle, nd s is te semiperimeter (lf of te perimeter). Use Hero s formul to find te re of tis tringle. Use te formul A 1 to ceck your nswer. 60 cm y eiter metod 11. Developing Proof Eplin wy must e 69 in te digrm t rigt. Becuse AOB is isosceles, ma 0 nd maob 140. mab 140 nd mcd 8. mac mbd ecuse prllel lines intercept congruent rcs on circle C D A F AF 10.5 cm CE 10. cm 1. As P moves from left to rigt long, wic of te following vlues cnges? E A. Te re of ABP P B. Te re of PDC C. Te re of trpezoid ABCD D D. ma mpcd mcpd E. None of tese A E A C BC 16.0 cm 15 cm 8 cm 17 cm 0 O 8 C D B B B Eercise 10 Tis eercise previews te eplortion following Lesson 8.7. [Ask] Wen is Hero s formul most elpful in finding te re of tringle? [wen te tringle s tree side lengts re given witout ny given ltitude] EXTENSIONS A. For Figure H in te investigtion, sow ow y reking te curve into more pieces, you will come closer to te ctul re under te curve. [Link] Tis is topic tt will e delt wit in integrl clculus. B. Hve model of uilding ville wit different surfces to e pinted, stined, singled, or tiled. Provide te cost of ec mteril nd sk te students to clculte te totl epense for ec jo. IMPROVING YOUR VISUAL THINKING SKILLS Four-Wy Split How would you divide tringle into four regions wit equl res? Tere re t lest si different wys it cn e done! Mke si copies of tringle nd try it. IMPROVING VISUAL THINKING SKILLS Encourge cretive tinking ere. Tis prolem cn e done in undreds of wys. One unusul wy if disconnected regions re llowed is to rerrnge pieces of te tringle to mke rectngle of te sme re, divide te resulting rectngle into four pieces of equl re, nd ten rerrnge tose pieces ck into tringle. LESSON 8.3 Are Prolems 437

20 USING YOUR ALGEBRA SKILLS 8 PLANNING ALGEBRA SKILLS USING 1 USING YOUR YOUR ALGEBRA SKILLS 1 18 USING YOUR ALGEBRA SKILLS 8 USING Y Products, Fctors, nd Qudrtic Equtions LESSON OUTLINE One dy: 5 min Emples 0 min Eercises MATERIALS grp pper lger tiles, optionl TEACHING Students etend te formul for te re of rectngle to fctoring lgeric epressions wit rectngle digrms. EXAMPLE A Encourge students to mnipulte lger tiles or rectngulr pieces of pper. [Alert] Students my mke ssumptions out ow mny smller pieces fit into lrger ones. For emple, tey my tink tt squre (representing ) equls si or seven of te smller rectngles (ec representing ). Point out tt sides of lengt cn e ligned only wit oter sides of lengt nd tt te sme olds for sides of lengt 1. Altoug te mnipultives migt e rrnged in vrious wys, tey will ll represent te sme fctoring. EXAMPLE A Solution Te re of rectngle is te product of two dimensions, te se nd eigt: A. So, you cn model te product of ny two numers or epressions s te re of rectngle. For emple, te rectngle t rigt s se nd eigt 3. Te re of te rectngle is mde of four smller rectngulr regions. Multiplying te se nd eigt of ec region gives res of,3,, nd 6. Comining te res gives te product: ( )( 3) 5 6 If n epression is te product of two oter epressions, you cn try to rrnge te pieces into rectngle nd find te fctors tt were multiplied. Sketc nd lel rectngle to fctor You my wnt to first sketc te pieces tt mke up te re. Visulize wys to rrnge ll of te pieces into rectngle. If you ve lger tiles or if you cut te pieces out of pper, you cn pysiclly eperiment wit different rrngements. Here is one wy to mke rectngle: Te se nd eigt give te fctors: 3 7 (3 1)( ) LESSON OBJECTIVES Use rectngle digrms to fctor qudrtic epressions Apply te zero-product property to solve qudrtic equtions COMMON CORE STANDARDS A.REI CHAPTER 8 Are

21 LGEBRA SKILLS 8 USING YOUR ALGEBRA SKILLS 8 USING YOUR ALGEBRA SKILLS 8 USING YO EXAMPLE B Solution Negtive vlues cn e represented y sutrction from side of rectngle. For emple, te sded rectngle t rigt s se 5 nd eigt 3, so its re cn e epressed s te product ( 5)( 3). 3 5 Te picture eqution elow sows wy ( 5)( 3) 15. Te overll rectngle s comined re 5. If you sutrct rectngulr region wit comined re 3 15, you re left wit te sded rectngle Wen n epression includes lrge numers or negtive numers, it is esier to use n strcted rectngle digrm to represent groups of pieces insted of drwing ec individul piece. Te rectngle t rigt sows te product ( 5)( 3). Add te res of te pieces to write te re s sum: ( 5)( 3) 15. Sketc nd lel rectngle digrm to fctor Drw rectngle digrm wit in te upper-left region nd 16 in te lowerrigt region. Te -region must ve dimensions y. Te remining two vlues must multiply to 16. Tey must lso dd to 6 ecuse tey re equl to te numer of -pieces, wic dd to 6. Troug some logic nd little tril nd error, you cn find te comintion of numers tt works. In tis cse, one of te numers must e negtive ecuse te product is negtive, nd te only fctor pirs for 16 re 1 16, 8, nd 4 4. Te rectngle digrm t rigt sows tt tt te solution is ( 8)( ). 15 Must multiply to Must dd to ( 8)( ) EXAMPLE B [Alert] Students migt e confused y te deprture from res, wic cnnot e negtive. Te confusion migt e eigtened if you refer to region s n re. Te sudivision of te rectngulr region is useful representtion, ut te res of te prts no longer represent te terms of te polynomil. As needed, sy tt tril nd error is perfectly vlid prolem-solving strtegy. EXAMPLE C To elp develop proof skills, [Ask] Wy re tere question mrks over te equl signs in te ceck? Te finding of solution is proofofte sttement If tis eqution s solution, ten 4 or 0. Te ceck is proof of te converse If 4 or 0, ten is solution to te eqution. But te second step of te ceck ssumes tt 4 nd 0 re solutions clim tt is still in question t tt point. SHARING IDEAS [Ask] Do you tink ll qudrtic epressions cn e fctored tis wy? Hve students eperiment t lest to te point of seeing tt frctions my e involved. Some students my recll te qudrtic formul. Closing te Lesson Summrize y ringing out te point tt mny qudrtic epressions, even wit negtive coefficients, cn e fctored using rectngle digrm. USING YOUR ALGEBRA SKILLS 8 Products,Fctors,nd Qudrtic Equtions 439

22 BUILDING UNDERSTANDING Te eercises llow furter prctice fctoring qudrtic epressions nd solving qudrtic equtions. ASSIGNING HOMEWORK Essentil 1 10, 14 17, 19 Performnce ssessment Portfolio 3 Group 11, 1, 18 1 Helping wit te Eercises ALGEBRA SKILLS 8 USING YOUR ALGEBRA SKILLS 8 USING YOUR ALGEBRA SKILLS 8 USING Y EXAMPLE C Solution Prolems involving re cn result in n eqution wit n term, clled qudrtic eqution. A qudrtic eqution cn e written in te form c 0, were,, nd c represent constnts. One metod for solving qudrtic eqution is to first fctor c. Te zero-product property sys tt if te product of two (or more) fctors is zero, ten t lest one of te fctors is zero: if 0, ten 0 or 0. Tis gives you wy to solve fctored qudrtic eqution. Solve Te originl eqution. Sutrct 4 from ot sides to set te eqution equl to zero. ( 4)( 6) 0 Fctor te left side. Use rectngle digrm if necessry. 4 0 or 6 0 Te zero-product property sys t lest one fctor is zero. 4 or 6 Solve ec eqution for. Ceck ot solutions: (4) (4)? 4 (6) (6)? 4 Sustitute 4 for, ten sustitute 6 for ? ? 4 Simplify following te order of opertions Bot solutions ceck (3)( 1) 3 1 EXERCISES In Eercises 1 6, sketc nd lel rectngle to find te product or to fctor te epression. As in Emple A, sow te individul pieces ( 5)( 1). ()( 7) 3. (3 )( 5) In Eercises 7 18, sketc nd lel rectngle digrm to find te product or to fctor te epression. As in Emple B, do not sow te individul pieces. 5. ( 5)( 3) ( 15)( 11) 8. (3 7)(4 5) 9. ( 4) or ( 4)( 4) 10. ( 5)( 5) 11. ( ) 1. ( )( ) ( 3)( 4) Eercises 9 1 Tese eercises review perfect squre trinomils nd te difference of two squres CHAPTER 8 Are

23 LGEBRA SKILLS 8 USING YOUR ALGEBRA SKILLS 8 USING YOUR ALGEBRA SKILLS 8 USING YO In Eercises 19, solve te qudrtic eqution ( 6) or 8 4 or 1 10 or 3. ( )( ) or 4 3 nd. 3. One se of n isosceles trpezoid is equl to te trpezoid s eigt nd te oter se is 4 feet longer tn te eigt. Te re of te trpezoid is 48 squre feet.. Let represent te trpezoid s eigt in feet. Sketc te trpezoid nd lel te eigt.. Write n epression tt includes for ec se of te trpezoid. Add tese to 4 your digrm. c. Write n eqution tt includes for te re of te trpezoid. IMPROVING YOUR REASONING SKILLS Tic-Tc-Squre Tis is one of mny vritions on tic-tc-toe. Insted of trying to get your X s or O s colliner, te oject is to get your X s or O s rrnged t te vertices of squre. Tere re mny wys to get four X s or O s rrnged t te vertices of squre. Two completed emple gmes re sown elow. Numers re used to sow te sequence of moves in ec gme. Answer te questions out ec of te four unfinised gmes. You cn identify possile move y te letter of te row nd numer of te column. For emple, te top-rigt squre in te grid is 5. O 6 O 10 Emple A Emple B O 9 O 3 X 8 X 6 O 5 X 8 O 1 O7 O 5 X 1 O 1 O 3 O 7 O 13 X 4 X X 6 O O 4 X 3 c X 4 X O 9 X 14 X 7 O 8 d O 17 O 11 X 10 X 1 X 18 X 5 e X 16 O 15 In tis gme, O went first nd won. In tis gme, X went first nd won. Were sould O ply to prevent X from winning? O 8 X 7 X 9 O 14 X 5 X 1 O O 10 O 1 X 1 X 3 c O 6 X 3 O 4 c O 1 O 3 X c X 13 X 5 O O 4 X 11 d d X 4 d O 8 X 7 e e e X 9 Eplin wy O s lst move, 4, ws mistke 1 [ ( 4)] 48 d. Solve te eqution in prt c to find te dimensions of te trpezoid. 8 or 6. Te eigt cnnot e negtive, so te only vlid solution is 6.Teeigtis6feet,oneseis 6 feet, nd te oter se is 10 feet. IMPROVING REASONING SKILLS Were sould O ply O 5 nd O 7 to gurntee win on O 9? Te lst move forces X to ply in c4. Tis cretes two squre options for X, c4-d-1-3 nd c4-c--4. O cn t lock ot of tese in one move, so X will win. 3. ; c1 Wic moves were forced? Wo sould win? Eplin ( 15)( 4) ( 1)( ) Eercises Refer students to teir work on Eercises 9 1. Eercise 0 [Alert] Some students my forget te step of mking one side of te eqution zero. [Ask] How cn you ceck your nswers? [Sustitute tem ck into te eqution nd see if ot sides simplify to te sme result.] Eercise 3 Severl representtions led to te sme eqution. [Alert] In 3d, solving te eqution is not enoug to find te dimensions of te trpezoid. Te negtive solution must e omitted, nd te lengts of te ses must e clculted. 4. X 7, X 9, O 10, X 11, X 13,nd O 14 were forced; X will win. See pge 84 for nswers to Eercises USING YOUR ALGEBRA SKILLS 8 Products,Fctors,nd Qudrtic Equtions 441

24 LESSON 8.4 PLANNING LESSON OUTLINE One dy: 15 min Investigtion 10 min Sring 10 min Closing 10 min Eercises MATERIALS construction tools clcultors Sketcpd ctivity Ares of Regulr Polygons, optionl TEACHING Te re of ny regulr polygon cn e determined y finding res of tringles. Te circles round te polygons in te student ook foresdow Lesson 8.5. One Step Cllenge students to find formul for te re of ny regulr n-gon (regulr polygon wit n sides). Some my devise different metods for polygons wit n even nd n odd numer of sides. In te unlikely event tt no group is tinking of dividing te polygon into congruent isosceles tringles, comment on te wkwrdness of one group s pproc nd plnt te ide of congruent isosceles tringles. INTRODUCTION If students found te re of Figure G in Lesson 8.3, [Ask] Wt were some of te wys you found tt re? Led te clss to gree tt te most generlizle metod of dividing ny regulr polygon is to prtition it into isosceles tringles rditing from te center. LESSON 8.4 If I d to live my life gin, I d mke te sme mistkes, only sooner. TALLULAH BANKHEAD LESSON OBJECTIVES Step 5 1 s Ares of Regulr Polygons You cn divide regulr polygon into congruent isosceles tringles y drwing segments from te center of te polygon to ec verte. Te center of te polygon is ctully te center of circumscried circle, so tese congruent segments re sometimes clled te rdii of regulr polygon. In tis investigtion you will divide regulr polygons into tringles. Ten you will write formul for te re of ny regulr polygon. Investigtion Are Formul for Regulr Polygons Consider regulr pentgon wit side lengt s, divided into congruent isosceles tringles. Ec tringle s se s nd eigt. Step 1 Wt is te re of one isosceles tringle in terms of nd s? 1 s Step Wt is te re of tis pentgon in terms of nd s? Step 3 Repet Steps 1 nd wit oter regulr polygons nd complete te tle elow. s Regulr egon Derive te formul for te re of regulr polygon Apply re formuls to solve prolems Lern new voculry Clrify pproimtion concepts Develop prolem-solving skills nd coopertive evior Te distnce ppers in te re formul for regulr polygon, nd it s specil nme potem. An potem of regulr polygon is perpendiculr segment from te center of te polygon s circumscried circle to side of te polygon. You my lso refer to te lengt of te segment s te potem. s Regulr eptgon Numer of sides n Are of regulr polygon 5 s 3s 7 s 4s 9 s 5s... 6s 1... sn s Regulr pentgon Apotem 44 CHAPTER 8 Are

25 Step 4 Wt is te perimeter of regulr polygon in terms of n nd s? Use your nswer to tis question nd your lst entry in te tle to stte your net conjecture. Regulr Polygon Are Conjecture A 1 sn A 1 P Te re of regulr polygon is given y te formul?or?,were A is te re, P is te perimeter, is te potem, s is te lengt of ec side, nd n is te numer of sides. C-79 [ELL] Add te following terms to te voculry disply: regulr polygon, potem, perimeter, pentgon, nd nongon. [Lnguge] Apotem is pronounced ['-pə-tem]. Guiding te Investigtion EXERCISES In Eercises 1 8, use te Regulr Polygon Are Conjecture to find te unknown lengt ccurte to te nerest unit, or te unknown re ccurte to te nerest squre unit. Recll tt te symol is used for mesurements or clcultions tt re pproimtions. 1. A? 09 cm.? 74 cm 3. P? 56 cm s 4 cm s cm 38.6 cm 4.9 cm A 19,887.5 cm A cm s 4. Regulr pentgon: 3cm nd s 4.4 cm, A? 5. Regulr nongon: 9.6 cm nd A 30.4 cm, P? 6. Regulr n-gon: 1 cm nd P 81.6 cm, A? 7. Find te pproimte perimeter of regulr polygon if 9m nd A 59. m. 8. Find te pproimte lengt of ec side of regulr n-gon if 80 feet, n 0, nd A 0,000 squre feet. 5 ft 9. Construction Drw segment 4 cm long. Use compss nd strigtedge to construct regulr egon wit sides congruent to tis segment. Use te Regulr Polygon Are Conjecture nd centimeter ruler to pproimte te egon s re. 4 cm 10. Drw regulr pentgon wit potem 4 cm. Use te Regulr Polygon Are Conjecture nd centimeter ruler to pproimte te pentgon s re. 58 cm 11. A squre is lso regulr polygon. How is te potem of squre relted to te side lengt? Sow tt te Regulr Polygon Are Conjecture simplifies to s for te re of squre. 1 s; A 1 sn 1 s s 4 s 1. Tecnology Use geometry softwre to construct circle. Inscrie pentgon tt looks regulr nd mesure its re. Now drg te vertices. How cn you drg te vertices to increse te re of te inscried pentgon? To decrese its re? Answers re on pge 444. Assessing Progress You cn ssess students understnding of regulr polygons, congruence, nd te re of figure s te sum of res of its prts. You cn lso ceck students ility to clculte res of tringles, use squre units in re mesurements, nd reson inductively. s 1 33 cm 490 cm 63 cm Closing te Lesson You will need Construction tools for Eercise 9 Geometry softwre for Eercises 1 nd 18 Reiterte te min point of tis lesson: Te re of regulr polygon is lf te product of te potem nd te perimeter. It s equivlent to 1 sn, were is te potem, s is ec side lengt, nd n is te numer of sides. s 57.6 m Step 1 [Alert] Some students my ve missed te ssumption tt te pentgon is regulr. Step 3 Groups my wnt to divide up te work. Cllenge students wo finis erly to rewrite te formul in terms of te polygon s perimeter. Ten suggest tt tey grp on teir clcultors tese ordered pirs for n-gons inscried in circles wit rdius 1: (n, re) [pproces ]; (n, ) [pproces 1]; (n, perimeter) [pproces ]. SHARING IDEAS First cll on students to fill in lnks in te tle. Ten ve students sow vriety of formuls, nd let tem find lgeric equivlents. [Ask] Wy do te isosceles tringles ll ve te sme re? How would you sow tt tey re congruent? [Two sides of ec tringle re congruent rdii of te circumscried circle. Te regulrity of te polygon ensures tt te tird sides re congruent, so use SSS. Or regulrity gives te congruence of te centrl ngles, so use SAS.] You cn link to Lesson 8.5 y wondering loud ow close te polygonl res re to te re of te circumscried circle. Elicit te ide tt, for lrge numer of sides, te potem is close to te rdius nd te perimeter is close to te circumference. You cn leve unnswered for now te question of finding te re of circle. LESSON 8.4 Ares of Regulr Polygons 443

26 BUILDING UNDERSTANDING Te eercises provide prctice in pplying te formuls for te re of regulr polygons. ASSIGNING HOMEWORK 13. Find te pproimte re of te 14. Find te pproimte re of te sded sded region of te regulr regulr egonl donut. Te potem nd octgon ROADSIGN. Te potem sides of te smller egon re lf s long s mesures 0 cm. Segment GI te potem nd sides of te lrge egon. mesures out 16.6 cm. 996 cm 6.9 cm nd r 8cm 497 cm G I r N S r Essentil 1 8, 15 Performnce ssessment 8 Portfolio 14 Group 9 13 Review 16 1 Alger, 3, 5, 7, 8, 11, review 15 17, 19 Helping wit te Eercises Eercise 3 If students re ving troule, sk wt te formuls re nd wt ppens wen te given mesurements re sustituted. Tey will e solving te eqution (38.6)P. Eercise 9 If students re stuck, [Ask] Didn t we see wy of constructing regulr egons ck in Cpter 0 or Cpter 1? [Lesson 0.3 nd Lesson 1.7] Eercise 10 Te instructions ere re to drw, not construct. 1. It is impossile to increse its re, ecuse regulr pentgon mimizes te re. Any drgging of te vertices decreses te re. (Susequent drgging to spce tem out more evenly cn increse te re gin, ut never eyond tt of te regulr pentgon.) Eercise 14 Students will feel est if, on teir own, tey cn get te ide of sutrcting te re of te smll egon from tt of te lrger. R Creer O Interior designers, unlike interior decortors, re concerned wit te lrger plnning nd tecnicl considertions of interiors, s well s wit style nd color selection. Tey ve n intuitive sense of sptil reltionsips. Tey prepre sketces, scedules, nd udgets for client pprovl nd inspect te site until te jo is complete. A D 15. Appliction An interior designer creted te kitcen pln sown. Te countertop will e constructed of colored concrete. Wt is its totl surfce re? If concrete countertops 1.5 inces tick cost $85 per squre foot, wt will e te totl cost of tis countertop? totl surfce re 13,680 in 95 ft ;cost $ Are is 0 squre units. y y 1_ 5 (, 6) 60 in. 48 in. 60 in. 36 in. 36 in. 17. Are is 36 squre units. y y 4_ in. 10 in. 138 in. 186 in. 4 in. 60 in. 48 in. 7 in. y 10 (6, 4) y _ CHAPTER 8 Are

27 Review In Eercises 16 nd 17, grp te two lines, ten find te re ounded y te -is, te y-is, nd ot lines. 16. y 1 5, y y 1 3 6, y Tecnology Construct tringle nd its tree medins. Compre te res of te si smll tringles tt te tree medins formed. Mke conjecture nd support it wit convincing rgument. 19. If te pttern continues, write n epression for te perimeter of te nt figure in te picture pttern. nw ny y y w y w y y w y w w w y w y w w y w y 0. GHJK is rectngle. Find te re FELA nd CDLB re prllelogrms. Find te of pentgon GHIJK. 504 cm re of te sded region. 840 cm A 3 cm K J B L 1 cm 18 cm 15 cm I 9 cm 6 cm 0 cm G 30 cm H 4 cm F 44 cm C E D EXTENSIONS A. Cllenge students to crete noter metod for rriving t te formul for te re of regulr polygon. Consider wys of dividing regulr polygon into isosceles tringles nd rerrnging te pieces into spes for wic students lredy know te re.use point t te center nd divide te polygon into n isosceles tringles, were is te potem, s is te lengt of ec side, nd n is te numer of sides in te regulr polygon. If te polygon s n odd numer of sides, te tringles cn e rrnged into trpezoid wit long se n 1 s, sort se n 1 s, eigt, nd re n 1 s n 1 s 1 ns. n 1 s IMPROVING YOUR VISUAL THINKING SKILLS Te Squred Squre Puzzle Te squre sown is clled squred squre. A squre 11 units on side is divided into 1 squres. Te re of squre X is 50, or 500, nd te re of squre Y is 4, or 16. Find te re of ec of te oter squres. A B X Y E C K I F Q J M L G N R O D S H P s If te polygon s n even numer of sides, te tringles cn e rrnged into prllelogrm wit se n s, eigt, nd re 1 ns. s s n n 1 s s n_ B. Use geometry softwre to demonstrte te Regulr Polygon Are Conjecture. See pge 84 for nswer to Eercise 18. IMPROVING VISUAL THINKING SKILLS Students my ve difficulty eginning tis puzzle, ut once tey get strted tey will find te rest mngele. If tey ve troule t te eginning, elp tem see tt tey know two tings out te sides of squres A nd B: te sum of teir mesures nd te difference. (Tey migt use lowercse letters for te sides of te squres.) No furter mnipultion of two vriles is needed. Tese re te res of te squres. A: 9 841; B: ; C: ; D: ; E: 5 65; F: 16 56; G: 18 34; H: 4 576; I: 9 81; J: 7 49; K: 15 5; L: 4; M: 17 89; N: 6 36; O: 11 11; P: ; Q: 35 15; R: 8 64; S: LESSON 8.4 Ares of Regulr Polygons 445

28 EXPLORATION PLANNING LESSON OUTLINE One dy: 35 min Investigtion 10 min Sring nd Closing MATERIALS squre dot pper or grp pper geoords, optionl Pick s Formul for Are (W), optionl Irregulr Spes (W), optionl TEACHING Ares of irregulr figures, polygonl nd curved, cn e estimted y plcing te figure on grid. One Step Direct te clss s ttention to polygons A G nd sk if tey cn find formul for te res in terms of te numers of dots on te interior. Elicit te ide tt te numer of dots on te oundry migt lso e needed. Cllenge students to work in groups to find formul, perps filling in tle suc s te one in Step 10. As you circulte, encourge systemtic dt gtering nd elp students focus on finding ptterns in rows nd columns of te tle. INTRODUCTION Polygons A nd B cn e sudivided into squre nd tringle(s), nd te res of tese figures cn ten e dded. Polygon C cn e seen s te result of removing tree tringles from rectngle wit re 1. [Alert] Students migt try to find te lengt of segment y counting te numer of dots on it rter tn te numer of units etween dots. LESSON OBJECTIVE Pick s Formul for Are You know ow to find te re of polygon regions, ut ow would you find te re of te dinosur footprint t rigt? You know ow to plce polygon on grid nd count squres to find te re. Aout undred yers go, Austrin mtemticin Georg Alender Pick ( ) discovered reltionsip, now known s Pick s formul, for finding te re of figures on squre dot grid. Let s strt y looking t polygons on squre dot grid. Te dots re clled lttice points. Let s count te lttice points in te interior of te polygon nd tose on its oundry nd compre our findings to te res of te polygon tt you get y dividing tem into rectngles nd tringles. Polygon A Polygon B Polygon C 5 oundry points 1 interior point Are Discover Pick s formul for finding te re of n irregulr figure drwn on grid 5 oundry points 0 interior points Are oundry points 5 interior points Are 1 (1.5 3 ) 5.5 How cn te oundry points nd interior points elp us find te re? An importnt tecnique for finding ptterns is to old one vrile constnt nd see wt ppens wit te oter vriles. Tt s wt you ll do in te net ctivity. In tis ctivity you will first investigte ptterns tt emerge wen you old te re constnt. Ten you will investigte ptterns tt emerge wen you old te numer of interior or oundry points constnt. Te gol is to find formul tt reltes te re of polygon to te numer of interior nd oundry points. 446 CHAPTER 8 Are

29 You will need squre dot pper or grp pper, or te workseets Pick s Formul for Are nd Irregulr Spes (optionl) geoords (optionl) Step 4 Answers will vry. Polygons wit 3 nd 4 re sown. Are 0.5 Are 1 Step 6 Answers will vry. Polygons wit 3 nd 4 re sown. Are 1.5 Are Activity Dinosur Footprints nd Oter Spes Step 1 Confirm tt ec polygon D troug G ove s re A 1. Step Step 3 Step 4 Let e te numer of oundry points nd i e te numer of interior points. Crete nd complete tle like tis one for polygons wit A 1. Study te tle for ptterns. Do you see reltionsip etween nd i? Grp te pirs (, i) from your tle nd lel ec point. Wt do you notice? Write n eqution tt fits te points. Now investigte wt ppens if you old te interior points constnt. Polygon B s 5, i 0, nd A 1.5. On squre dot pper drw oter polygons wit no interior points. Clculte te re of ec polygon. Step 5 Copy nd complete tis tle for polygons wit i 0. Step 6 Step 7 Step 8 Step 8 Answers will vry. Polygons wit Step 9 i nd i 3 re sown. Are.5 Are 3.5 Step i 5 E F G D Numer of oundry points () Numer of interior points (i) E Polygon A s 5, i 1, nd A.5. On squre dot pper drw oter polygons wit ectly one interior point. Clculte te re of ec polygon. Mke tle like te one in Step 5, ut for polygons wit i 1. Wen you old te interior points constnt, wt ppens to te re ec time one oundry point is dded? Now investigte wt ppens if you old te oundry points constnt. F Polygon C s 3, i 5, nd A 5.5. On squre dot pper drw oter polygons wit ectly tree oundry points. Clculte te re of ec polygon. Copy nd complete tis tle for polygons wit 3. Wen you old te oundry points constnt, wt ppens to te re ec time one interior point is dded? Te re increses y 1. D G D E F G Numer of oundry points () Are Numer of interior points (i) Are Step 7 Te re increses y 0.5. Boundry points () Are G uiding te Activity Geoords migt elp wit tis eplortion. Students cn sow teir figures for Steps 4, 6, nd 8 on te Pick s Formul for Are workseet. You migt do tis ctivity s jigsw. Assign Steps 1 3 to one or two groups, Steps 4 nd 5 to noter group, Steps 6 nd 7 to tird group or pir of groups, nd Steps 8 nd 9 to noter group. Ten form new groups to tckle Steps Step 3 Students sould lel te es. As needed, [Ask] Wt do you wnt to figure out to determine te eqution of te line? Te eqution is i Step 11 Students my wonder wt to write s coefficient of i, climing tt it s no numer in front of it in te formul tey derived. Help tem see tt te coefficient is 1. Step 1 You migt nd out te workseet Irregulr Spes. Pick s formul is ccurte for polygons wit vertices on grid points. It is not very ccurte for tese irregulr spes. One metod for finding te re of te irregulr spes is to connect te dots of te grid wit segments s close s possile to te perimeter of te spe. Tis cretes polygonl region wit pproimtely te sme re s te irregulr spe. Students cn ten count te oundry points nd interior points of te polygonl figure nd use Pick s formul to pproimte te re of te irregulr spe. Te points lie on te line i EXPLORATION Pick s Formul for Are 447

30 Step 1 smple nswers: dinosur: 71 units,or out 160 in ;lef:69units,or out 177 cm ;Tes:84units, or out 30,400 mi SHARING IDEAS [Ask] Are te res clculted in Step 1 precise? [All of tem re pproimte. Pick s formul gives n ect re only of polygon wose vertices re t grid points, so te res clculted ere re te result of pproimting te re of polygons.] Wt would ppen if you moved one spe on te grid? [Te numers of oundry nd interior points could cnge, ut te re covered y te spe would not cnge.] You migt mention tt Pick s formul is one of mny metods for pproimting res of irregulr figures. Some metods use counting, wile oters use proility or clculus. To get t te mening of te coefficients in te formul, especilly 1,you migt suggest tt students strt y drwing te smllest possile tringle on te grid wit re 1 nd ten see ow te re cnges s tey dd one grid point t time to te oundry witout introducing ny interior points. Tey cn ten eperiment wit introducing interior points one t time. Anoter wy of tinking out it is tt ec grid point in te interior represents n entire unit squre of re, weres ec grid point on te oundry represents lf of unit squre. Assessing Progress You cn ssess students understnding of re nd re conservtion nd teir skill t gtering dt systemticlly nd resoning inductively. Closing te Lesson Step 10 Step 11 Step 1 Dinosur foot To find Pick s formul, it cn e elpful to orgnize ll your dt in one plce. Copy nd complete te tle elow. Ceck te ptterns y drwing more polygons on squre dot pper. Generlize te formul to find te re if you know te numer of oundry points nd te numer of interior points. Copy nd complete te conjecture. Pick s formul is useful for pproimting te re of n irregulrly sped region. 6 in. You cn find or pproimte te re of n irregulr spe y first putting it on grid nd ten using Pick s formul, A 1 i 1. Te formul will determine te ect re of ny polygon wose vertices re t grid points. Are (s function of nd i) Numer of oundry points () Numer of interior points (i) Pick s Formul If A is te re of polygon wose vertices re lttice points, is te numer of lttice points on te oundry of te polygon, nd i is te numer of interior lttice points, ten A?? i?. 1,1,1 Use Pick s formul to find te pproimte res of tese irregulr spes. 8 cm Mple lef 300 mi Tes 448 CHAPTER 8 Are

31 LESSON 8.5 Te moon is drem of te sun. PAUL KLEE You will need compss scissors Step 1 Step Step 3 Ares of Circles So fr, you ve discovered te formuls for te res of vrious polygons. In tis lesson you ll discover te formul for te re of circle. Most of te spes you ve investigted in tis cpter could e divided into rectngles or tringles. Cn circle e divided into rectngles or tringles? Not ectly, ut in tis investigtion you will see n interesting wy to tink out te re of circle. Investigtion Are Formul for Circles Circles do not ve strigt sides like polygons do.however,te re of circle cn e rerrnged. Let s investigte. Use your compss to mke lrge circle. Cut out te circulr region. Fold te circulr region in lf. Fold it in lf second time, ten tird time nd fourt time. Unfold your circle nd cut it long te folds into 16 wedges. Arrnge te wedges in row, lternting te tips up nd down to form spe tt resemles prllelogrm. LESSON 8.5 PLANNING LESSON OUTLINE One dy: 0 min Investigtion 10 min Sring nd Emples 5min Closing 10 min Eercises MATERIALS construction tools clcultors scissors tpe or removle glue stick, optionl Sketcpd ctivity Ares of Circles, optionl TEACHING Step 4 COMMON CORE STANDARDS G.GMD.1 G.C.5 If you cut te circle into more wedges, you could rerrnge tese tinner wedges to look even more like rectngle, wit fewer umps. You would not lose or gin ny re in tis cnge, so te re of tis new rectngle, skimming off te umps s you mesure its lengt, would e closer to te re of te originl circle. If you could cut infinitely mny wedges, you d ctully ve rectngle wit smoot sides. Wt would its se lengt e? Wt would its eigt e in terms of C, te circumference of te circle? Te rdius of te originl circle is r nd te circumference is r. Give te se nd te eigt of rectngle mde of circle cut into infinitely mny wedges. Find its re in terms of r. Stte your net conjecture. LESSON OBJECTIVES Derive te formul for te re of circle Apply re formuls to solve prolems Develop coopertive evior Te re of circle cn e epressed in terms of its circumference nd terefore in terms of. [ELL] Review circle terms suc s rdius, dimeter, nd circumference. Guiding te Investigtion One Step Ask students to derive formul for te re of circle. Ask ny students wo rememer formul to give justifiction. Some my divide te circle into pieces. Ask students wo re stumped weter tey ve seen someting recently out relting circles to inscried polygons. You migt demonstrte te investigtion s students follow long. Step 1 Mke te circle lrge. Step 3 Students cn tpe or glue (removle glue sticks work well) te wedges onto seet of pper LESSON 8.5 Ares of Circles 449

32 Guiding te Investigtion (continued) nd lel te se nd eigt of te prllelogrm in terms of te circumference nd rdius of te circle. Circle Are Conjecture Te re of circle is given y te formul?,were A is te re nd r is te rdius of te circle. A r C-80 SHARING IDEAS Hve students present wtever vriety of sttements tey ve come up wit for te Circle Are Conjecture. Some my ve derived 1 Cr, wile oters my ve found d 4 or te more fmilir r. [Ask] How cn you rememer tt te re is r nd te circumference is r? Let students propose vrious wys. One wy uses dimensionl nlysis; r s te sme liner units s r, weres r s squre units. [Ask] Does it mke sense for to e in formul for te re of circle? Students my see wy circle s re out tree times te re of squre uilt on its rdius, ut wy sould tt fctor e? Tis fct eluded te clssicl Greek mtemticins, despite teir knowledge tt te rtio of circumference to dimeter ws constnt. Assessing Progress You cn ssess students understnding of re, of prllelogrms nd ow to find teir res, of rdius, nd of circumference s constnt multiple of dimeter. EXAMPLE A Te dimeter of te smll pie nd te rdius of te lrge pie re given. Students my discuss weter lrger refers to dimeter, perimeter, re, or volume. [It is ment to refer to re.] Te question How muc lrger? cn refer to eiter difference or rtio. Te solution gives ot. Te rtio of lrge re to smll re is not 10 to 8 (15%) ut rter 100 to 64 (156%). [Alert] Students my not understnd wen to epress teir nswers using te symol nd EXAMPLE A Solution EXAMPLE B How do you use tis new conjecture? Let s look t few emples. Te smll pple pie s dimeter of 8 inces, nd te lrge cerry pie s rdius of 5 inces. How muc lrger is te lrge pie? First, find ec re. Smll pie Lrge pie A r A r (4) (5) (16) (5) Te lrge pie is 78.5 in, nd te smll pie is 50. in. Te difference in re is out 8.3 squre inces. So te lrge pie is more tn 50% lrger tn te smll pie, ssuming tey ve te sme tickness. Notice tt we used 3.14 s n pproimte vlue for. If te re of te circle t rigt is 56 m, wt is te circumference of te circle? Use te re to find te rdius, ten use te rdius to find te circumference. Solution 56 m A r C r 56 r (16) 56 r 3 r m Te circumference is 3 wen to use n pproimtion for. Tey sould use te symol until te end nd ten pproimte if tey need deciml vlue. EXAMPLE B Tis emple demonstrtes te convenience of ving results in terms of nd not pproimting until te end. If students rec for teir clcultors in te first step, tey ll miss te esy cncelltion of nd perps fil to rrive t te ect vlue for r. meters, or pproimtely meters. Closing te Lesson Te re of circle cn e found in severl wys, ll involving eiter directly or indirectly. Te most common formul is A r.te formul A 1 Cr comes eiter from wedges or from seeing te circle s te limit of inscried regulr polygons. [Link] Looking for te limit s te numer of divisions pproces infinity is te sis of clculus. 450 CHAPTER 8 Are

33 EXERCISES Use te Circle Are Conjecture to solve for te unknown mesures in Eercises 1 8. Leve your nswers in terms of, unless te prolem sks for n pproimtion. For pproimtions, use te key on your clcultor. 1. If r 3 in., A?. 9 in. If r 7 cm, A?. 49 cm 3. If r 0.5 m, A?. 0.8 m 4. If A 9 cm,ten r?. 3 cm 7. If C 1 in., ten A?. 36 in 8. If C 314 m, ten A? m 9. Wt is te re of te sded region 10. Wt is te re of te sded region etween te circle nd te rectngle? etween te circle nd te tringle? y (3, 4) (5, 0) ( 8, 6) y You will need (8, 6) Geometry softwre for Eercise If A 3 in,ten r?. 3 in. 6. If A m,ten r?. 0.5 m 5 48, or out 30.5 squre units , or out 186 squre units (10, 0) BUILDING UNDERSTANDING As needed, you migt ve students work on severl eercises, especilly some in wic tey need to pproimte te vlue of, nd ten sre teir results. ASSIGNING HOMEWORK Essentil 1 14 Performnce ssessment 13 Portfolio 1 Journl 16 Group 15 Review 17 Alger review 4 10, Sketc nd lel circle wit n re of 34 cm. Be sure to lel te lengt of te rdius. 1. Appliction Te rotting sprinkler rms in te poto t rigt re ll 16 meters long. Wt is te re of ec circulr frm? Epress your nswer to te nerest squre meter. 804 m 13. Appliction A smll college TV sttion cn rodcst its progrmming to ouseolds witin rdius of 60 kilometers. How mny squre kilometers of viewing re does te sttion rec? Epress your nswer to te nerest squre kilometer. 11,310 km 14. Smpson s dog, Cecil, is tied to post y cin 7 meters long. How muc ply re does Cecil ve? Epress your nswer to te nerest squre meter. 154 m 15. Appliction A muscle s strengt is proportionl to its cross-sectionl re. If te cross section of one muscle is circulr region wit rdius of 3 cm, nd te cross section of second, identicl type of muscle is circulr region wit rdius of 6 cm, ow mny times stronger is te second muscle? 4 times r 18 cm Cmpion weigt lifter Sory Jimenez etends rells weiging lmost doule er own ody weigt. Eercises 1 15 If students re ving difficulty, encourge tem to drw pictures. Eercise 15 [Lnguge] A cross section of treedimensionl oject is te intersection of tt oject wit plne perpendiculr to te is. If you cut tin slice of n oject, you cn see cross section. You my need to eplin wt it mens for te strengt of one to e proportionl to its crosssectionl re. Tis ide will e revisited in Cpter 11 in te Eplortion Wy Elepnts Hve Big Ers. Helping wit te Eercises None of te nswers involving mesurement cn e ect. However, wen is left in te nswer, n equl sign sows tt no rounding s een done. Approimte nswers (wit n pproimtion sign, ) re used for nswers tt ve een rounded. Becuse is irrtionl, clcultions wit lwys require rounding. Eercise 5 Students my e unfmilir wit rdicls nd clim tt tere s no numer wose squre is 3. Ask weter tere s wy to get te rdius from te coefficient 9 in Eercise 4 nd weter tt sme metod migt pply to te coefficient 3 in tis eercise. It s etter to leve te nswer in terms of 3, just s wit, ut ifstudents wnt n pproimte vlue tey cn use clcultor. Eercise 6 Here, finding n pproimte squre root wit clcultor is more importnt tn in Eercise 5. Eercise 9 If students re stuck, sk weter ny metods tey ve used efore could e elpful ere. Tey ve often used sutrction. LESSON 8.5 Ares of Circles 451

34 19. Te tringles ve equl re wen te point is t te intersection of te two digonls. Tere is no oter loction t wic ll four tringles ve equl re. Eercise Becuse te liquid fills lf of te continer in te originl position, it will lso fill lf of te continer in te new position. Terefore te eigt of te liquid will e lf te eigt of te continer in te new position: 1 (4) 1 cm.. 1 cm 4 cm Wt would e good pproimtion for te re of regulr 100-gon inscried in circle wit rdius r? Eplin your resoning. A r ecuse te 100-gon lmost completely fills te circle. Review 17. A? 456 cm A? 36 ft 19 cm 4 cm 19. Tecnology Construct prllelogrm nd point in its interior. Construct segments from tis point to ec verte, forming four tringles. Mesure te re of ec tringle. Move te point to find loction were ll four tringles ve equl re. Is tere more tn one suc loction? Eplin your findings. 0. Developing Proof Eplin wy must e Developing Proof Wt s wrong wit tis picture? mde C B 4 D 15 ft 38 8 ft 10 ft 9 ft 6 cm 18 cm A E Te 6-y-18-y-4 cm cler plstic seled continer is resting on cylinder. It is prtilly filled wit liquid, s sown. Sketc te continer resting on its smllest fce. Sow te liquid level in tis position. 6 cm 18 cm 4 cm IMPROVING YOUR VISUAL THINKING SKILLS Rndom Points Wt is te proility of rndomly selecting from te 3-y-3 grid t rigt tree points tt form te vertices of n isosceles tringle? IMPROVING VISUAL THINKING SKILLS To encourge students wo need elp, [Ask] How mny isosceles tringles ve teir verte ngle t tis point? [3 wit verte ngle t ec corner, 4 wit verte ngle t ec side midpoint, nd 8 wit verte ngle in te center; ] Students cn use symmetry to consider only four points. Fmilirity wit counting comintions will elp tem see tt tere re 84 possile wys to select tree points on te grid ; 36 out of 84 gives proility of CHAPTER 8 Are

35 LESSON 8.6 Cut my pie into four pieces I don t tink I could et eigt. YOGI BERRA Any Wy You Slice It In Lesson 8.5, you discovered formul for clculting te re of circle. Wit te elp of your visul tinking nd prolem-solving skills, you cn clculte te res of different sections of circle. Its mkers climed tis ws te world s lrgest slice of pizz. If you cut slice of pizz, ec slice would proly e sector of circle. If you could mke only one strigt cut wit your knife, your slice would e segment of circle. If you don t like te crust, you d cut out te center of te pizz; te crust spe tt would remin is clled n nnulus. LESSON 8.6 PLANNING LESSON OUTLINE One dy: 10 min Investigtion or Emples 5min 5min Sring Closing 5 min Eercises MATERIALS none TEACHING Sector of circle Segment of circle Annulus r r r A sector COMMON CORE STANDARDS G.C.5 A sector of circle is te region etween two rdii nd n rc of te circle. A segment of circle is te region etween cord nd n rc of te circle. An nnulus is te region etween two concentric circles. Picture equtions re elpful wen you try to visulize te res of tese regions. Te picture equtions elow sow you ow to find te re of sector of circle, te re of segment of circle, nd te re of n nnulus. r 360 r r 1_ A segment LESSON OBJECTIVES R r R r R r A nnulus Discover formuls nd metods for clculting te re of nnuluses, sectors, nd segments of circles Lern new voculry Develop prolem-solving skills nd coopertive evior One Step Refer students to te investigtion in Lesson 8.5, in wic tey folded circle nd cut out wedges. Ask wt prt of te circle s re ws in ec wedge. Now you ll consider te reverse question. Pose tis prolem, dpted from Eercise 13: Suppose te pointed ngle of tis slice of pizz is 36, te long sides ve lengt 0 feet, nd one cn of tomto suce will cover 3 squre feet of pizz. How mny cns would you need to cover tis slice? As needed, encourge students to drw pictures. [Ask] Hve you decided weter or not it would elp to find te re of te wole circulr pizz? Is tere ny wy to determine wt prt tis slice is of te wole? Introduce te terms sector, segment, nd nnulus during Sring, nd ten consider ow to clculte te re of te ltter two regions. INTRODUCTION [ELL] Add te new terms sector of circle, segment of circle, nd nnulus to te voculry disply wit digrms. LESSON 8.6 Any Wy You Slice It 453

36 [Alert] As you discuss circle segment, students my e tinking of cut troug te center, semicircle,wic is ot segment nd sector. Be sure students understnding includes ll segments. Bot segment nd sector of circle cn e greter tn semicircle. [Lnguge] Annulus is Ltin for ring. As you discuss wit students te definition of nnulus, ceck tt tey know wt concentric mens. EXAMPLE A If you ve pysicl model, students would proly enefit from seeing or even ndling it. Te picture eqution epresses te re of sector in terms of te rc mesure of te sector, weres tis emple works wit only te centrl ngle. Ask ow te numers re relted. Becuse tey re te sme, you cn [Ask] Wtprtoftereofte 45 circle is in te sector? EXAMPLE B Emples A nd B ppel to different lerning styles. Emple A is intentionlly presented in two columns for nlyticl, liner lerners, wile Emple B, in prgrp form, is more verl. You migt lso wnt to sow Emple B in two columns. Some students my need to see more detils in te derivtion: A 3 60 r 1 0 (6) 1 (6)(6) cm 18 cm (9 18) cm 10.3 cm A result involving my not e so useful if you ve concrete reson to know te re. Most students will enefit from lerning to estimte. [Ask] To te nerest wole numer, wt is te vlue of 9 minus 18? [ is out 3, so 9 18 is out 9.] Or students cn use clcultor nd round te nswer to te pproprite numer of significnt digits. EXAMPLE A Solution EXAMPLE B Solution EXAMPLE C Solution Find te re of te sded sector. 45 Te sector is 3 6, 0 or 1,ofte 8 circle. A sector 3 60 r Te re formul for sector (0) Sustitute r 0 nd Reduce te frction nd squre Te re is 50 cm. Find te re of te sded segment. Multiply. According to te picture eqution on pge 453, te re of segment is equivlent to te re of te sector minus te re of te tringle. You cn use te metod in Emple A to find tt te re of te sector is cm, or 9 cm. Te re of te tringle is 1 6 6, or 18 cm. So te re of te segment is 9 18 cm. Te sded re is 14 cm,nd te rdius is 6 cm. Find. Te sector s re is 36 0 of te circle s re, wic is Te centrl ngle mesures 140. EXAMPLE C As needed, remind students tt te mesure of te centrl ngle is te sme s te mesure of te rc. You my wnt to point out tt te formul eing used is A 3 60 r. For students wit weker lger skills, you my need to eplin tt te second step multiplies ot sides of te eqution 360 y 3 6, or tt tis step my e seen s two steps: multiplying ot sides y 360 nd dividing ot sides y cm 45 0 cm 6 cm SHARING IDEAS You my ve Sring fter te one-step investigtion or fter students ve worked on some of te eercises. Ask wt te key ide is in finding te re of sector. Mny students ve difficulty wit rtios like te one emedded ere. Empsize tt tey re just finding wt prt of te re of te wole circle is in te sector. Tt prt is te sme s te prt of 360 tt s in te ngle mesure of te sector. 454 CHAPTER 8 Are

37 EXERCISES In Eercises 1 8, find te re of te sded region. Te rdius of ec circle is r. If two circles re sown, r is te rdius of te smller circle nd R is te rdius of te lrger circle. 1. r 6 cm 6 cm. r 8 cm 64 3 cm 3. r 16 cm 19 cm r cm ( ) cm 5. r 8 cm (48 3) cm 6. R 7 cm r 4 cm 33 cm 7. r cm 1 cm 8. R 1 cm 9. Te sded re is 1 cm. r 9 cm 10 5 cm Find r. 6 cm r 10. Te sded re is 11. Te sded re is 1. Te sded re is 10 cm. 3 cm. Find r. 10 cm, nd te Te rdius of te lrge circle is 7 cm rdius is 4 cm. Find. 10 cm, nd te rdius of te 75 smll circle is 8 cm. Find. 100 r R 60 r R You will need r r Construction tools for Eercise 16 Sring Ides (continued) Students encountered similr resoning wen clculting rc lengt. Te lengt of te rc ws te sme prt of te circumference s te mesure of te centrl ngle ws of 360. Ask ow segments of circles re relted to line segments. Tey re different ecuse line segments re one-dimensionl ojects nd segments of circles ve two dimensions. On te oter nd, line segment cross te circle (tt is, cord) cuts off segment of te circle. [Ask] Are sectors nd segments of circle prts of te circle? [Actully, tey re not; tey re prt of te interior of te circle.] Ask weter students tink tey need to memorize formuls for te re of sector,segment,nd nnulus. It s muc more importnt tt tey understnd te mening of te terms nd ow to find te re from te re of circle. BUILDING UNDERSTANDING 13. Appliction Suppose te pizz slice in te poto t te eginning of tis lesson is sector wit 36 rc, nd te pizz s rdius of 0 ft. If one cn of tomto suce will cover 3 ft of pizz, ow mny cns would you need to cover tis slice? 4 Assessing Progress Troug students work on te eercises, you cn ssess teir understnding of circle circumference nd re. Closing te Lesson Te re of sector of circle is te sme prt of te re of te wole circle s te mesure of te centrl ngle of te sector is of 360. Te re of segment of circle cn e found y sutrcting te re of tringle from tt of te corresponding sector. Te re of n nnulus is te difference in res of te outside nd inside circles. Becuse no emple of n nnulus is given, you my wnt to work troug Eercises 6, 7, 8, 10, or 1. Students my find tese eercises difficult. Be prepred to give plenty of support to keep down te frustrtion level. ASSIGNING HOMEWORK Essentil 1 1 Performnce ssessment 13, 14 Portfolio 13, 14 Group 15, 16 Review 17 3 Alger review 9 14 MATERIALS Eercise 15 (T), optionl Eercise 17 (T), optionl LESSON 8.6 Any Wy You Slice It 455

38 Helping wit te Eercises Eercises 1 8 As needed, suggest tt students focus on te unsded s well s te sded regions. You migt work emples to demonstrte te re of te sded regions nd ten during omework discussion sk for te re of te unsded regions. Eercise 3 Ask frustrted students weter tis eercise is like Eercise, to elp tem see tt tey need to sutrct te given ngle mesure from 360. Eercise 4 As needed, [Ask] Wt s needed to find te re of segment? Cn you sketc picture eqution? Eercise 8 Students migt find te re of te wole nnulus first, or tey migt find te prtil res of te circles nd ten sutrct. Tis is good eercise to sow tt mny prolems ve multiple pproces. Cllenge students to find multiple pproces to prolems. Eercises 9 1 Tese four eercises my e te most cllenging of te lesson. Eercise 9 As needed, sk students weter tey cn epress te re, 1, in terms of r nd write n eqution. Eercise 11 If students re ving difficulty, wonder loud weter it would elp to use te fct tt te sded re is te sme prt of te wole re s te ngle mesure is of 360. Tis oservtion leds to n epression. Eercise 1 As needed, sk wt migt e first step. Students cn find te re of te wole nnulus first nd ten consider wt prt of tt re 10 is. 14. Utopi Prk s just instlled circulr fountin 8 meters in dimeter. Te Prk Committee wnts to pve 1.5-meter-wide pt round te fountin. If pving costs $10 per squre meter, find te cost to te nerest dollr of te pved pt round te fountin. $448 Mtemtics Attempts to solve te fmous prolem of squring circle finding squre wit te sme re s given circle led to te cretion of some specil spes mde up of prts of circles. Te digrms elow re sed on some tt Leonrdo d Vinci sketced wile ttempting to solve tis prolem. 15. Te illustrtions elow demonstrte ow to find rectngle wit te sme re s te sded figure. In series of digrms, demonstrte ow to find rectngle wit te sme re s te sded figure... c. d. 16. Construction Reverse te process you used in Eercise 15. On grp pper, drw 1-y-6 rectngle. Use your compss to divide it into t lest four prts, ten rerrnge te prts into new curved figure. Drw its outline on grp pper. Eercise 15 It migt elp students to descrie in words ow te curved region is cnged into rectngulr region in te emple. [Contet] Leonrdo d Vinci ( ) ws n Itlin pinter, drftsmn, sculptor, rcitect, nd engineer. His tlents were so numerous nd so diverse tt mny feel e epitomizes te Renissnce idel. 16. smple nswer: Tis circulr fountin t te Cteu de Villndry in Loire Vlley, Frnce, sres center wit te circulr pt round it. How mny concentric rcs nd circles do you see in te picture? See pge 84 for nswers to Eercise CHAPTER 8 Are

39 Review Ec set of circles is eternlly tngent. Wt is te re of te sded region in ec figure? Wt percentge of te re of te squre is te re of te circle or circles in ec figure? All given mesurements re in centimeters c. 1 d. 1 (144 36) cm ; 78.54% (144 36) cm ; 78.54% (144 36) cm ; 78.54% (144 36) cm ; 78.54% 18. Te eigt of trpezoid is 15 m nd te midsegment is 3 m. Wt is te re of te trpezoid? 480 m 19. CE, BH, nd AG re ltitudes. Find AB nd AG. AB 17.0 cm, AG 6.6 cm A G C 5.0 cm 7.1 cm E H cm 1.0 cm In Eercises 0 3, identify ec sttement s true or flse. If true, eplin wy. If flse, give counteremple. 0. If te rc of circle mesures 90 nd s n rc lengt of 4 cm, ten te rdius of te circle is 48 cm. 90 True. If r, ten r 48 cm. 1. If te mesure of ec eterior ngle of regulr polygon is 4, ten te polygon s 15 sides. True. If 3 60 n 4, ten n 15.. If te digonls of prllelogrm isect its ngles, ten te prllelogrm is squre. Flse. It could e romus. 3. If two sides of tringle mesure 5 cm nd 30 cm, ten te tird side must e greter tn 5 cm, ut less tn 55 cm. true; Tringle Inequlity Conjecture IMPROVING YOUR REASONING SKILLS Code Equtions Ec code elow uses te first letters of words tt will mke te eqution true. For emple, 1M Y is n revition of te eqution 1 Monts Yer. Find te missing words in ec code D n AA of n IRT. 7 S of H 3. 90D ec A of R 4. 5 D in P B Eercise 17 Students my e surprised to discover tt prts d ll ve te sme nswer. [Ask] Wy re te res te sme? Algericlly, n 1 n is 36 no mtter wt n is. More insigtfully, perps, wen you divide te liner dimensions of figure y n, you divide its re y n,so n copies of te reduced figure will ve te sme re s te originl. EXTENSIONS A. Hve students reserc te istory of te prolem of squring (qudrture of) te circle. B. Pose tis prolem: A friend tells you se s found noter wy to find te re of n nnulus. First find te verge of te circumferences of te two circles tt form te nnulus. Ten multiply tt verge y te tickness of te nnulus (te difference etween te rdii) to find te re. Is your friend rigt? Drw picture nd use lger to sow wy tis metod does or doesn t work. Tis metod works, ecuse R r (R r) (R r)(r r) R r R r. R r R r IMPROVING REASONING SKILLS Degrees in n Acute Angle of n Isosceles Rigt Tringle. 7 Sides of Heptgon Degrees in ec Angle of Rectngle 4. 5 Digonls in Pentgon C. Te re of trpezoid cn lso e seen s te product of its midsegment nd its eigt. Ask students to eplin ow tis ide reltes to te re of portion of n nnulus. R r is te verge of te two circumferences. LESSON 8.6 Any Wy You Slice It 457

40 EXPLORATION PLANNING LESSON OUTLINE One dy: 45 min Activity MATERIALS rulers pennies dimes protrctors grp pper TEACHING Some pplictions of geometry to proility cn e unepected. G uiding te Activity Te Spe of Tings Step 1 You my need to empsize tt te computer is picking point nywere in te grid, not necessrily in one of te spes. Students cn find te res of te squre nd te pentgon y sutrcting res of tringles from te re of n enclosing rectngle. 15 Step 1 P(trpezoid) ; P(pentgon) ; P(squre) ; 4 P(tringle) Step Students migt interpret te word qudrilterl to refer to ll of te spes insted of only to te trpezoid nd squre. Tey cn eiter dd te two proilities or dd te two res nd divide y 144; te ltter metod is preferle, ecuse it s ect. Te proility tt te computer-selected point is outside ll of te spes cn e found from eiter te sum of te You will need grp pper ruler penny dime LESSON OBJECTIVES Use res of circles nd polygons to clculte proility Crete geometric models to nswer proility questions Geometric Proility II You lredy know tt proility vlue is numer etween 0 nd 1 tt tells you ow likely someting is to occur. For emple, wen you roll die, tree of te si rolls nmely,, 3, nd 5 re prime numers. Becuse ec roll s te sme cnce of occurring, P(prime numer) 3 6,or 1. In some situtions, proility depends on re. For emple, suppose meteorite is eded towrd Ert. If out 1 3 of Ert s surfce is lnd, te proility tt te meteorite will it lnd is out 1, 3 wile te proility it will it wter is out. 3 Becuse Alsk s greter re tn Vermont, te proility te meteorite will lnd in Alsk is greter tn te proility it will lnd in Vermont. If you knew te res of tese two sttes nd te surfce re of Ert, ow could you clculte te proilities tt te meteorite would lnd in ec stte? Activity Were te Cips Fll In tis ctivity you will solve severl proility prolems tt involve re. Te Spe of Tings At ec level of computer gme, you must coose one of severl spes on coordinte grid. Te computer ten rndomly selects point nywere on te grid. If te point is outside your spe, you move to te net level. If te point is on or inside your spe, you lose te gme. On te first level, trpezoid, Spe Vertices pentgon, squre, nd tringle Trpezoid (1, 1), (8, 1), (7, 9), (4, 9) re displyed on grid tt goes from 0 to 1 on ot es. Te Pentgon (3, 1), (4, 4), (6, 4), (9, ), (7, 0) tle elow gives te vertices of Squre (0, 6), (3, 9), (6, 6), (3, 3) te spes. Tringle (11, 0), (7, 4), (11, 1) 458 CHAPTER 8 Are

41 B B I I Step Step 1 33 P(qudrilterl) ; P(outside Step ll spes) Step 3 Step 3 1 P(tringle) Step 4 Step 6 Togeter, Step 5 te sectors mke 60 Step 6 up 3 6, 0 or 1 6,ofte totl re. So, P(Bonus) 1 Step Step 8 Step 9 Yes; te Step 9 drt is no longer eqully likely to it ny spot on te ord, so you cn no longer compute proilities y simply finding nd dividing res. Step 10 For ec spe, clculte te proility te computer will coose point on or inside tt spe. Epress ec proility to tree deciml plces. Wt is te proility te computer will coose point tt is on or inside qudrilterl? Wt is te proility it will coose point tt is outside ll of te spes? If you coose tringle, wt is te proility you will move to te net level? Wic spe sould you coose to ve te est cnce of moving to te net level? Wy? Trpezoid; te proility tt te computer will select point inside te trpezoid is less tn te proility for ny oter spe. Rigt on Trget You re plying crnivl gme in wic you must trow one drt t te ord sown t rigt. Te score for ec region is sown on te ord. If your drt lnds in Bonus section, your score is tripled. Te more points you get, te etter your prize will e. Te rdii of te circles from te inside to te outside re 4 in., 8 in., 1 in., nd 16 in. Assume your im is not very good, so te drt will it rndom spot. If you miss te ord completely, you get to trow gin. Wt is te proility your drt will lnd in te red region? Blue region? Compute te proility your drt will lnd in Bonus section. (Te centrl ngle mesure for ec Bonus section is 30.) If you score 90 points, you will win te grnd prize, gint stuffed emu. Wt is te proility you will win te grnd prize? If you score ectly 30 points, you win n I Crnivls sell cp. Wt is te proility you will win te cp? Now imgine you ve een prcticing your drt gme, nd your im s improved. Would your nswers to Steps 5 8 cnge? Eplin. Te Coin Toss You own smll cfe tt is populr wit te mtemticins in te neigorood. You devise gme in wic te customer flips coin onto red-nd-wite ceckered tleclot wit 1-inc squres. If it lnds completely witin squre, te customer wins nd doesn t ve to py te ill. If it lnds toucing or crossing te oundry of squre, te customer loses. Te Coin Toss Step 10 Give students grp pper, or ve tem drw teir own tleclot, for pysicl trils. For coin to e completely inside squre, te distnce from its center to side of te squre must e greter tn its rdius, r. Te center of te coin must e witin squre of side lengt 1 r, or 1 d, centered in te 1-inc squre. A penny s dimeter of out 3 4 in., so te penny would ve to lnd witin squre wit side lengt 1 3 4,or 1 4 in. Bonus Bonus Assuming te coin stys on te tle, wt is te proility of te customer winning y flipping penny? A dime? (Hint: Were must te center of te coin lnd in order to win?) I N I N LIBERTY G O D G O D W E W E 1996 T R U S T T R U S T R E E L L T R T Y Y 1996 IN GOD WE TRUST 1996 A dime s dimeter of out in., so it would ve to lnd witin squre of side lengt , 5 or 1 6 in. r r r r Center of coin must lnd in sded squre. 1 in. Step (continued) proilities or te sum of te res. Te former gives sligtly different result due to round-off error. As wen tey re working wit, for more ccurte results students sould wit until te end to do ny rounding. Step 3 As needed, remind students tt tey move to te net level if te rndomly selected point is outside te cosen spe. Rigt on Trget re of red Step 5 P(red) r e of trget ; re of lue , so P(lue) re of lue re of trget Step 7 Students could compute te proility s P(red nd Bonus) 1 P(red) P(Bonus) Step 7 Te drt must lnd in te prt of te red section witin Bonus section. 1 6 (16) 8 3, P(90 points) Step 8 To score 30 points, te drt must lnd eiter in te prt of te red section tt is not in Bonus section, 5 6 (16) 40 3, or in te prt of te green section tt is in Bonus section, 1 6 (144 64) Step 8 P(30 points) EXPLORATION Geometric Proility II 459

42 Step 10 P(win wit penny) 1 re of center squre 1 6 re of 1-inc squre P(win wit dime) 5 re of center squre 56 re of 1-inc squre Step 11 If te coin must lnd in red squre, te proilities determined in Step 10 re multiplied y 1. On Different Note Step 13 Before students egin on tis step, you migt give tem few minutes to try to figure out tis proility. Ten tey migt e more impressed y te re process, ecuse te sitution pprently s noting to do wit geometry. Steps 13, 14 Pollione s times (min) y Ties Rigoletto s times (min) Step 14 P(Rigoletto) re of grid elow line re of grid [Contet] Te fictitious oper singers re crcters in fmous opers y Verdi nd Bellini. Assessing Progress You cn ssess students understnding of proility s rtio. Step 11 P(win wit penny) Step 11 If te customer wins only if te coin flls witin red squre, wt is te proility of winning wit penny? A dime? Step 1 Suppose te gme is lwys plyed wit penny, nd customer wins if te P(win wit dime) penny lnds completely inside ny squre (red or wite). If your dily proceeds 1 verge $300, out ow muc will te gme cost you per dy? ($300) 6 $18.75 Step 13 Step 14 DIFFERENT DICE On Different Note Two oper strs Rigoletto nd Pollione re uditioning for prt in n upcoming production. Becuse te singers ve similr qulifictions, te director decides to ve contest to see wic mn cn old note te longest. Rigoletto s een known to old note for ny lengt of time from 6 to 9 minutes. Pollione s een known to old note for ny lengt of time from 5 to 7 minutes. Drw rectngulr grid in wic te ottom side represents te rnge of times for Rigoletto nd te left side represents te rnge of times for Pollione. Ec point in te rectngle represents one possile outcome of te contest. On your grid, mrk ll te points tt represent tie. Use your digrm to find te proility tt Rigoletto will win te contest. Understnding proility cn improve your cnces of winning gme. If you roll pir of stndrd 6-sided dice, re you more likely to roll sum of 6 or 1? It s firly common to roll sum of 6, ecuse mny comintions of two dice dd up to 6. But 1 is only possile if you roll 6 on ec die. If you rolled pir of stndrd 6-sided dice tousnds of times, nd recorded te numer of times you got ec sum, te istogrm would resemle te one t rigt. If you rolled tem fr fewer times, your istogrm would look more irregulr. Would te distriution e different if you used different dice? Wt if one die d odd numers nd te oter d even numers? Wt if you used 8-sided dice? Wt if you rolled tree 6-sided dice insted of two? Coose one of tese scenrios or one tt you find interesting to investigte. Mke your dice nd roll tem 0 times. Predict wt te grp will look like if you roll te dice 100 times, ten ceck your prediction. Your project sould include Your dice. Histogrms of your eperimentl dt, nd your predictions nd conclusions. You cn use Ftom to simulte rel events, suc s rolling te dice tt you ve designed. You cn otin te results of undreds of events very quickly, nd use Ftom s grping cpilities to mke istogrm sowing te distriution of different outcomes. Supporting te You migt connect tis project wit te Cpter 10 eplortion on Pltonic solids. A student wo s completed te project wit Ftom cn demonstrte te results dynmiclly. OUTCOMES Predictions mention possile outcomes nd most likely outcomes. Irregulrities in weigt nd spe of dice re minimized. Te istogrm s rnge reflects te minimum nd mimum possile outcomes nd s te pproimte correct spe. Simultion results rougly mtc te teoreticl proility. Report mkes connection etween te numer of trils nd ow closely te outcomes mtc te epected vlues. Unusul dice re eplored. Report compres teoreticl outcomes for severl kinds of dice. 460 CHAPTER 8 Are

43 LESSON 8.7 No pessimist ever discovered te secrets of te strs, or siled to n uncrted lnd, or opened new doorwy for te umn spirit. HELEN KELLER Surfce Are In Lesson 8.3, you clculted te surfce res of wlls nd roofs. But not ll uilding surfces re rectngulr. How would you clculte te mount of glss necessry to cover pyrmid-sped uilding? Or te numer of tiles needed to cover cone-sped roof? In tis lesson you will lern ow to find te surfce res of prisms, pyrmids, cylinders, nd cones. Te surfce re of ec of tese solids is te sum of te res of ll te fces or surfces tt enclose te solid. For prisms nd pyrmids, te fces include te solid s ses nd its lterl fces. In prism, te ses re two congruent polygons nd te lterl fces re rectngles or oter prllelogrms. In pyrmid, te se cn e ny polygon. Te lterl fces re tringles. Bses Lterl fce Bse Lterl fce LESSON 8.7 PLANNING LESSON OUTLINE One dy: 5min Emples 0 min Investigtion 5min 5min Sring Closing 10 min Eercises MATERIALS clcultors prisms, pyrmids, nd oter solids Tis glss pyrmid ws designed y I. M. Pei for te entrnce of te Louvre museum in Pris, Frnce. Tis skyscrper in Cicgo, Illinois, is n emple of prism. A cone is prt of te roof design of tis Victorin ouse in Msscusetts. LESSON OBJECTIVES Review terminology for solids Tis stone tower is cylinder on top of lrger cylinder. Discover metods for finding te surfce res of solids Prctice visul tinking in tree dimensions Develop prolem-solving skills nd coopertive evior TEACHING It s possile to find te surfce re of mny solids, suc s regulr prisms, pyrmids, cylinders, nd cones. We re ssuming ll te solids re rigt prisms, pyrmids,cylinders,or cones. Hving rel ojects cn elp students visuliztion. Boes cn e cut prt. Cns cn e covered wit pper, wic cn ten e removed nd flttened. You migt remind students tt tey ve seen nets erlier in te course. Empsize tt finding surfce re is ctully finding te re of te net. One Step Remind students of wt prism, cylinder, pyrmid, nd cone re, nd eplin wt teir lterl surfce re is. Ten sk te clss to do two tings: find formul for te lterl surfce re of prism wose se is regulr polygon, ten see wt ppens to tt formul s te numer of sides increses nd s one se srinks to point. As you circulte, encourge groups to tink out cutting te figures to ly tem flt, to consider one kind of cnge t time, nd to divide LESSON 8.7 Surfce Are 461

44 One Step (continued) up te lor. During Sring, introduce te term slnt eigt for one lengt students re using, nd sk students ow to find te totl surfce re of ec figure, including te res of te ses. INTRODUCTION To focus ttention on te surfce re of solids, you migt sk students: If rick s si fces, ow mny fces does lf rick ve? At first one or two students my lug t te spoof on mindless lgeric tinking, ut if you persist tey my see tt splitting rick in lf digonlly cn result in lf rick wit only five fces. Ask weter tere re oter possile nswers, nd leve te question open to encourge tree-dimensionl visuliztion. [ELL] Enlrge te digrms on pge 461 tt sow te prism nd pyrmid nd relted terms. Crete similr digrms for cylinder nd cone. EXAMPLE A [Alert] Some students, tinking of volume formul, my sy tt tey know simple formul to find te surfce re. Tey my not relize tt surfce re is te numer of squre units tt it will tke to cover te two-dimensionl surfce of tree-dimensionl figure. EXAMPLE A Solution EXAMPLE B Solution To find te surfce res of prisms nd pyrmids, follow tese steps. Steps for Finding Surfce Are 1. Drw nd lel ec fce of te solid s if you d cut te solid prt long its edges nd lid it flt. Lel te dimensions.. Clculte te re of ec fce. If some fces re identicl, you only need to find te re of one. 3. Find te totl re of ll te fces. Find te surfce re of te rectngulr prism. 3 m Drw nd lel te two congruent ses, nd te four lterl fces unfolded into one rectngle. Ten find te res of ll te rectngulr fces. 8 6 m 8 m 6 Top Front Side Bck Side surfce re (se re) (lterl surfce re) (6 8) 3( ) (48) 3(8) Te surfce re of te prism is 180 m. Find te surfce re of te cylinder. Imgine cutting prt te cylinder. Te two ses re circulr regions, so you need to find te res of two circles. Tink of te lterl surfce s wrpper. Slice it nd ly it flt to get rectngulr region. You ll need te re of tis rectngle. Te eigt of te rectngle is te eigt of te cylinder. Te se of te rectngle is te circumference of te circulr se. 6 Bottom Tese sipping continers re rectngulr prisms. 1 in. 10 in. 46 CHAPTER 8 Are

45 1 Top 5 Bses C r Lterl surfce Bottom surfce re r ( r) 5 ( 5) Te surfce re of te cylinder is out 534 in. 5 EXAMPLE B Students cn mke te sides of cylinders out of pper, wit no overlps, nd ten see ow to find te lterl surfce re fter unrolling te pper. Slnt Heigt [Alert] Students my ve difficulty differentiting etween te eigt (perpendiculr to te se), te slnt eigt (n ltitude of fce), nd te lengt of n edge of pyrmid. Hving models ville cn elp. Tis ice crem plnt in Burlington, Vermont, uses cylindricl continers for its milk nd crem. Tese conservtories in Edmonton, Cnd, re glss pyrmids. Te surfce re of pyrmid is te re of te se plus te res of te tringulr fces. Te eigt of ec tringulr lterl fce is clled te slnt eigt. Te slnt eigt is leled l nd te pyrmid eigt is leled. Heigt In te investigtion you ll find out ow to clculte te surfce re of pyrmid wit regulr polygon se. l Slnt eigt LESSON 8.7 Surfce Are 463

46 Guiding Investigtion 1 Steps 4, 5 Students my need ints to use te distriutive property to get A 1 n(l ) nd to recognize tt n P. Guiding Investigtion Step [Alert] You my need to elp students see tt s te numer of tringles increses, te lterl surfce of te pyrmid egins to look like te lterl surfce of cone. Terefore, 1 Pl ecomes 1 Cl, ecuse te perimeter ecomes circle. Step 3 Some students migt go on to write te epression 1 Pl r s 1 (r)l r, wic simplifies to r(r l). SHARING IDEAS As students present ides, encourge creful use of terminology. Tese figures re clled solids, toug tey re not filled in. Tt is, te solid is only te surfce, two-dimensionl figure wose re you cn find. Te surfce my contin twodimensionl fces surrounded y one-dimensionl edges. [Ask] Wen we refer to prticulr kind of prism, suc s pentgonl prism, wt do we men? [Te spe of te ses is eing specified.] Ask weter te formuls for lterl surfce re re relted. Elicit te ide tt tey re ll te sme: If you consider pyrmid or cone s ving two ses, one of perimeter zero, ten te lterl surfce is te slnt eigt times te verge of te two se perimeters. Tis migt remind students of te reltionsip mong re formuls for prllelogrms, trpezoids, nd tringles. [Ask] If trpezoid is te intermedite step etween prllelogrm nd tringle, is tere n nlogy in tree dimensions etween prism nd pyrmid wit te sme se? Step 1 1 l Step 8 1 l, n 1 l Step 3 1 P or 1 n Step 4 1 n(l ) Step 5 1 P(l ) l Step 1 Step Step 3 Step 4 Step 5 Investigtion 1 Surfce Are of Regulr Pyrmid You cn cut nd unfold te surfce of regulr pyrmid into tese spes. l Is te roof of tis uilding in Ksn, Irn, cone or pyrmid? Wt mkes it rd to tell? Wt is te re of ec lterl fce? Wt is te totl lterl surfce re? Wt is te totl lterl surfce re for ny pyrmid wit regulr n-gon se? Wt is te re of te se for ny regulr n-gon pyrmid? Use your epressions from Steps nd 3 to write formul for te surfce re of regulr n-gon pyrmid in terms of n, se lengt, slnt eigt l, nd potem. Write noter epression for te surfce re of regulr n-gon pyrmid in terms of eigt l, potem, nd perimeter of te se, P. l 1_ P You cn find te surfce re of cone using metod similr to te one you used to find te surfce re of pyrmid. l 464 CHAPTER 8 Are

47 Step 1 r Step r l r l ; l r l lr Step 1 Step Step 3 Step 3 r(r l) Tis potogrp of Siou tepees ws tken round 190 in Nort or Sout Dkot. Is tepee sped more like cone or pyrmid? Investigtion Surfce Are of Cone As te numer of fces of pyrmid increses, it egins to look like cone. You cn tink of te lterl surfce s mny smll tringles or s sector of circle. r l Wt is te re of te se? Wt is te lterl surfce re in terms of l nd r? Wt portion of te circle is te sector? Wt is te re of te sector? Write te formul for te surfce re of cone. l r r r l Sring Ides (continued) [Te intermedite step is clled frustum.] You migt mention tt tere re wys to find te surfce re of olique nd nonregulr solids s well. Students will study te surfce re of speres in Cpter 10. [Ask] Wt kinds of units do surfce re mesures ve? [Tey re mesures of re, so tey ve squre units.] Remind students tt in Lesson 8.6 tey found te re of sector of circle y considering wt prt it ws of wole circle. Now, to find te lterl surfce re of cone, tey tougt of cutting te cone nd lying out te surfce into sector. Wonder loud weter te re of tis sector cn e found in te sme wy. We don t know te ngle or te mesure of te rc, ut we do know te rc lengt, wic is r. Te wole circle would ve circumference l, so te re of te sector is r l times te re of te wole circle, wic is l. Tis product simplifies to lr, s found in Investigtion. EXAMPLE C Find te totl surfce re of te cone. If you feel students need more prctice in finding te surfce res of cones, proceed to Emple C. Oterwise, close te lesson nd egin on te eercises. EXAMPLE C Solution SA rl r ()(5)(10) (5) Te surfce re of te cone is out 36 cm. 5 cm 10 cm Students migt use ny of te formuls tey know for te surfce re of cone. Closing te Lesson Reiterte te min ides of tis lesson: Te surfce re of ny prism, pyrmid, cylinder, or cone is its lterl surfce re plus te re(s) of its se(s). Te lterl surfce re is te slnt eigt times te verge of te se perimeters, considering pyrmids nd cones to ve one se of perimeter zero. (Te perimeter of circle is its circumference.) LESSON 8.7 Surfce Are 465

48 BUILDING UNDERSTANDING Te eercises give prctice in finding surfce re. ASSIGNING HOMEWORK Essentil 1 10 Performnce ssessment 1 Portfolio 10 Journl 11 Group 13 Review Alger review 1, 16, 18 MATERIALS Eercise 1 (T), optionl Helping wit te Eercises EXERCISES In Eercises 1 10, find te surfce re of ec solid. All qudrilterls re rectngles, nd ll given mesurements re in centimeters. Round your nswers to te nerest 0.1 cm cm 4070 cm 6 16 cm cm Te se is regulr egon wit 8. Te se is regulr pentgon wit potem potem 1.1, side s 14, nd 11 nd side s 16. Ec lterl edge eigt cm t 17, nd te eigt of fce l cm You will need cm cm Construction tools for Eercise Eercises 1 8 As needed, encourge students to cut te solid into te vrious spes s illustrted in te emples. Ten clculte te re of ec plne figure. t l s s 9. D 8, d 4, l 8, w 4, 10, d 4 d w 39.1 cm d D cm l 11. Eplin ow you would find te surfce re of tis oelisk. re of squre 4 re of trpezoid 4 re of tringle 466 CHAPTER 8 Are

49 1. Appliction Cludette nd Mrie re plnning to pint te eterior wlls of teir country frmouse (ll verticl surfces) nd to put new cedr singles on te roof. Te pint tey like est costs $5 per gllon nd covers 50 squre feet per gllon. Te wood singles cost $65 per undle, nd ec undle covers 100 squre feet. How muc will tis ome improvement cost? All mesurements re in feet. $ smple nswer: smple tiling End view Construction Te spes of te spinning dises in te poto re clled frustums of cones. Tink of tem s cones wit teir tops cut off. Use your construction tools to drw pieces tt you cn cut out nd tpe togeter to form frustum of cone. Review 14. Use ptty pper, templtes, or pttern locks to crete / /4 4 tiling. 15. Trce te figure t rigt. Find te lettered ngle mesures nd rc mesures. 75, 75, c 30, d 60, e 150, f 30 A Sri Lnkn dncer lnces nd spins pltes. d 150 c e f EXTENSION / /4 4 In tis lesson, students found te surfce re of prism, pyrmid, nd cylinder witout using surfce re formuls. Tey used re formuls for polygons nd circles to find te res of te different surfces tt mde up ec solid nd ten dded tose res to find te totl surfce re.for te cone,tey derived formul for te lterl surfce re. Ask students to derive single formul for te totl surfce re of ec solid in te emples nd investigtions, using lger to mke te formuls s simple s possile. (In oter words, sk tem to use s few vriles s possile nd comine s mny terms s possile.) Ask tem to write nd present emples of teir formul. [prism: lw l w or (lw l w); squre-sed pyrmid: l or ( l); cylinder: r r or r(r ); cone: r rl or r(r l)] LESSON 8.7 Surfce Are 467

50 Appliction Suppose circulr rnc wit rdius of 3 km were divided into 16 congruent sectors. In one-yer cycle, ow long would te cttle grze in ec sector? Wt would e te re of ec sector? Aout 3 dys. Ec sector is out km. History In 179, visiting Europens presented orses nd cttle to Hwii s King Kmeme I. Cttle rncing soon developed wen Meicn vqueros cme to Hwii to trin Hwiins in rncing. Tody, Hwiin cttle rncing is ig usiness. Grzing geometry is used on Hwii s Ku Rnc. Rncers divide te grzing re into sectors. Te cows re rotted troug ec sector in turn. By te time tey return to te first sector, te grss s grown ck nd te cycle repets , 50, c 80, d 100, e 80, f 100, g 80, 80, k 80, m 0, n 80. Eplntions will vry. Smple eplntion: Te ngle wit mesure d corresponds to te ngle forming liner pir wit g. Becuse d 100, y te Prllel Lines Conjecture, te ngle djcent to g mesures 100, nd y te Liner Pir Conjecture, g 80. Te ngle wit mesure f corresponds to te ngle mesuring 100, so f 100. Te ngles mesuring g nd k re te se ngles of n isosceles tringle, so y te Isosceles Tringle Conjecture, k Developing Proof Trce te figure t rigt. Find te lettered ngle mesures. Eplin ow you determined mesures f nd k. 18. If te pttern of locks continues, wt will e te surfce re of te 50t solid in te pttern? (Every edge of ec lock s lengt 1 unit.) 398 squre units f g m k n 50 c e d IMPROVING YOUR VISUAL THINKING SKILLS Moving Coins Crete tringle of coins similr to te one sown. How cn you move ectly tree coins so tt te tringle is pointing down rter tn up? Wen you ve found solution, use digrm to eplin it. c d e f g i j IMPROVING VISUAL THINKING SKILLS If students re stuck, you migt wonder loud wt cnging ec row to te longest row would men for te oter rows. g c j d e f i 468 CHAPTER 8 Are

51 EXPLORATION Step 1 Step Step 3 Step 4 Step 5 Step 7 Te re of qudrilterl (rectngle) is equl to te product of Step 6 te mens of te lengts of opposite Step 7 sides. Alterntive Are Formuls In ncient Egypt, wen te yerly floods of te Nile River receded, te river often followed different course, so te spe of frmers fields long te nks could cnge from yer to yer. Officils ten needed to mesure property res in order to keep records nd clculte tes. Prtly to keep trck of lnd nd finnces, ncient Egyptins developed some of te erliest mtemtics. Historins elieve tt ncient Egyptin t ssessors used tis formul to find te re of ny qudrilterl: A 1 ( c) 1 ( d) were,, c, nd d re te lengts, in consecutive order, of te figure s four sides. In tis ctivity you will tke closer look t tis ncient Egyptin formul, nd noter formul clled Hero s formul, nmed fter Hero of Alendri. Activity Clculting Are in Ancient Egypt Investigte te ncient Egyptin formul for qudrilterls. Construct qudrilterl nd its interior. Cnge te lels of te sides to,, c, nd d, consecutively. Mesure te lengts of te sides nd use te Sketcpd clcultor to find te re ccording to te ncient Egyptin formul. Select te polygon interior nd mesure its re. How does te re given y te formul compre to te ctul re? Is te ncient Egyptin formul correct? no Does te ncient Egyptin formul lwys fvor eiter te t collector or te lndowner, or does it fvor one in some cses nd te oter in oter d cses? Eplin. Te formul fvors te t collector or is fir if te qudrilterl is rectngle. Descrie te qudrilterls for wic te formul works ccurtely. For wt kinds of qudrilterls is it sligtly inccurte? Very inccurte? Te formul works only for rectngles. Stte te ncient Egyptin formul in words, using te word men. LESSON OBJECTIVES Use geometry softwre to eplore re formuls Investigte Hero s formul for te re of tringles Eplore te Egyptin formul for te re of qudrilterl c PLANNING LESSON OUTLINE One dy: 35 min Activity 5min 5min MATERIALS Sring Closing Te Geometer s Sketcpd TEACHING Students investigte proposed formuls for te res of ll qudrilterls nd for te res of ll tringles. INTRODUCTION [Contet] Egypt d government, courts of lw, nd systems of ttion s erly s 3000 B.C.E. G uiding te Activity Step 1 To investigte n ritrry qudrilterl, students must construct one tt s no specil crcteristics. Tey cn djust qudrilterls ppropritely using Sketcpd. Step 7 Help students see tt tis formul migt mke intuitive sense to someone tinking of sligt devitions from rectngles. Step 8 If students re ving difficulty wit ccurcy, encourge tem to rek up te formul into pieces. Tey certinly won t wnt to reenter c ec time, so tey cn clculte tis quntity first. Tey migt lso wnt to consider te squre root of te product s product of t lest two squre roots. Hero s formul pplies to ny tringle. EXPLORATION Alterntive Are Formuls 469

52 [Contet] Hero, lso known s Heron, (c C.E.) my ve een lecturer t te Museum in Alendri. SHARING IDEAS Focus on wy te ncient Egyptin formul olds for rectngles ut not qudrilterls in generl. For emple, te re formul for trpezoids involves mens. [Ask] Wy doesn t te proposed formul pply to trpezoids? [Te men of te nonses would ve to e te eigt, ut ot nonses re longer tn n ltitude.] Te sme resoning cn e used to eplin wy te formul doesn t pply to ny prllelogrms oter tn rectngles: Te sides not considered ses re longer tn n ltitude. [Ask] Wen migt Hero s formul e useful? [wen te sides of te tringle re known ut no ltitude is known] You migt suggest tt students try etending Hero s formul to qudrilterls; in wt cses, if ny, is it ccurte? Assessing Progress Ceck students understnding of specil kinds of qudrilterls, teir ierrcy, nd teir res. Step 8 Step 9 According to Hero s formul, if s is lf te perimeter of tringle wit side lengts,, nd c, te re A is given y te formul A s(s )(s )(s ) c Use Sketcpd to investigte Hero s formul. Construct tringle nd its interior. Lel te sides,, nd c, nd use te Sketcpd clcultor to find te tringle s re ccording to Hero. Compre te result to te mesured re of te tringle. Does Hero s formul work for ll tringles? Hero s formul lwys works. Devise wy of clculting te re of ny qudrilterl. Use Sketcpd to test your metod. One possile metod: Divide te qudrilterl into two tringles, ten find te re of ec tringle. IMPROVING YOUR VISUAL THINKING SKILLS Cover te Squre Trce ec digrm elow onto noter seet of pper. Cut out te four tringles in ec of te two Cut out te smll squre nd te four smll equl squres nd rrnge tem to tringles from te squre on leg EF nd ectly cover te lrge squre. rrnge tem to ectly cover te lrge squre. A B C Arrnge te eigt tringles to fit in ere. Two squres wit res nd y re divided into te five regions s sown. Cut out te five regions nd rrnge tem to ectly cover lrger squre wit n re of z. y D F Two squres ve een divided into tree rigt tringles nd two qudrilterls. Cut out te five regions nd rrnge tem to ectly cover lrger squre. E Arrnge te five pieces to fit in ere. Closing te Lesson Te ncient Egyptin formul for te re of qudrilterl pplies only to rectngles, ut Hero s formul for te re of tringle pplies to ll tringles. y z IMPROVING VISUAL THINKING SKILLS Tis ctivity elps prepre students for te Pytgoren Teorem in Cpter 9. Assign it nd ve students sre teir results in clss. z y y z 470 CHAPTER 8 Are

53 EW CHAPTER CHAPTER 11 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHA 8 REVIEW EXERCISES You sould know re formuls for rectngles, prllelogrms, tringles,trpezoids,regulr polygons,nd circles.you sould lso e le to sow were tese formuls come from nd ow tey re relted to one noter. Most importntly, you sould e le to pply tem to solve prcticl prolems involving re, including te surfce res of solid figures. Wt occuptions cn you list tt use re formuls? Wen you use re formuls for rel-world pplictions, you ve to consider units of mesurement nd ccurcy. Sould you use inces, feet, centimeters, meters, or some oter unit? If you work wit circle or regulr polygon, is your nswer ect or n pproimtion? For Eercises 1 10, mtc te re formul wit te sded re. 1. A B (prllelogrm) A. B. C.. A A A 0.5d 1 d E (kite) D. E. F. 5. A 0.5P F (regulr polygon) 6. A r D (circle) 7. A 36 r 0 J (sector) 8. A R r I (nnulus) G. H. I. J. 9. SA rl r 10. SA rl r A (tringle) C (trpezoid) G (cylinder) H (cone) For Eercises 11 13, illustrte ec term Answers re on pge Apotem 1. Annulus 13. Sector of circle For Eercises 14 16, drw digrm nd eplin in prgrp ow you derived te re formul for ec figure. 14. Prllelogrm 15. Trpezoid 16. Circle Reviewing (continued) Ten imgine letting te numer of sides ecome infinite so tt te figure ecomes circle. [Ask] Wt s its re? Convert te re lgericlly to te stndrd formul. Ten consider piece of te circle, sector wit centrl ngle mesure 60. [Ask] Wt s its re? Find te re of te ssocited segment. Going ck to te squre, imgine moving it strigt out from plne to form squre prism. Anlyze its surfce re. Add n edge nd more edges to mke regulr n-gon prisms, nd nlyze teir surfce res. Srink te outer n-gon to mke frustum. [Ask] Wt s its surfce re? Srink te top ll te wy to point. [Ask] Wt s te surfce re of te pyrmid? Mke one more trip ck to te squre. Turn it into regulr n-gon, nd move it out to mke prism. Now, let te numer of edges get lrger to pproc cylinder. [Ask] Wt s its surfce re? Srink te outer circle to mke frustum nd ten cone, nd emine teir surfce res. CHAPTER REVIEW 8 PLANNING LESSON OUTLINE First dy: 15 min Reviewing 30 min Eercises Second dy: 10 min Reviewing 0 min Eercises 15 min Student self-ssessment REVIEWING You migt do quick dynmic review using geometry softwre or drwing on n ersle surfce. Strt wit squre, nd [Ask] Wt s its re, nd wy? Discuss squre units. Epnd te squre to rectngle, nd discuss its re in terms of squre units. Adjust te sides to mke trpezoid, nd discuss ow te squre units now ve to e cut prt ut te re is still te numer of units tt fit into te spce. Srink one se down to point, nd find te re of te resulting tringle. Go ck to te squre nd drw in digonls. Sow tt te re of te originl squre is lf te product of te lengts of te digonls. Now sift one edge to mke romus. [Ask] Hs te re cnged? Wt s te re of romus in terms of its digonls? Finlly, move one verte of te romus long digonl to mke kite. [Ask] Wt s te re of te kite? Return to squre, nd divide it into tringles nd find its re in terms of its perimeter. Increse te numer of sides to mke regulr pentgon. [Ask] Wt s its re in terms of its perimeter? Generlize to regulr n-gon. CHAPTER 8 REVIEW 471

54 ASSIGNING HOMEWORK You my wnt to ve students work on te eercises over most of two clss periods. Most students will enefit from doing some of te eercises in group nd some eercises individully. MATERIALS grp nd dot pper EW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPT Solve for te unknown mesures in Eercises All mesurements re in centimeters. 17. A? 800 cm 18. A? cm 19. A? 60 cm or out 36 R cm s 41.6 r 0 40 s R r Helping wit te Eercises Eercises 1 10 Tese eercises re quick, ut tey could e reviewed more tn once. Eercises Wit sketces, tese eercises sould ecome prt of ec student s noteook Apotem 0. A 576 cm 1. A 576 cm. A 16 cm? 3 cm d 1? 3 cm 13 cm 9 cm? 15 cm C 18 cm 4. A 576 cm 5. A sector 16 cm A? 81 cm Te circumference is?. mfan? cm A d 1 r N F In Eercises 6 8, find te sded re to te nerest 0.1 cm. In Eercises 7 nd 8, te qudrilterl is squre nd ll rcs re rcs of circle of rdius 6 cm Smple nswer: Construct n ltitude from te verte of n otuse ngle to te se. Cut off te rigt tringle nd move it to te opposite side, forming rectngle. Becuse te prllelogrm s re sn t cnged, its re equls te re of te rectngle. Becuse te re of te rectngle is given y te formul A, te re of te prllelogrm is lso given y A. 8 cm cm 15. Smple nswer: Mke copy of te trpezoid nd put te two copies togeter to form prllelogrm wit se 1 nd eigt. Tus te re of one trpezoid is given y te formul A cm 30.9 cm 16. Smple nswer: Cut circulr region into lrge numer of wedges nd rrnge tem into spe tt resemles rectngle. Te se lengt of tis rectngle is r nd te eigt is r, so its re is r. Tus te re of circle is given y te formul A r. r 1 r 47 CHAPTER 8 Are

55 CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 Eercise 3 Plcing C t (8, 8) or (8, 8) gives prllelogrms ABDC or ACBD, rter tn In Eercises 9 31, find te surfce re of ec prism or pyrmid. All given mesurements re in centimeters. All qudrilterls re rectngles, unless oterwise leled Te se is trpezoid ABCD. All tree prllelogrms ve te sme re. 3. Are is 11 squre units. y D (6, 8) C (0, 8) cm cm 300 cm A (0, 0) B (14, 0) For Eercises 3 nd 33, plot te vertices of ec figure on grp pper, ten find its re. 3. Prllelogrm ABCD wit A(0, 0), B(14, 0), nd D(6, 8) 33. Qudrilterl FOUR wit F(0, 0), O(4, 3), U(9, 5), nd R(4, 15) 34. Te sum of te lengts of te two ses of trpezoid is cm, nd its re is 66 cm.wt is te eigt of te trpezoid? 6 cm 35. Find te re of regulr pentgon to te nerest tent of squre centimeter if te potem mesures out 6.9 cm nd ec side mesures 10 cm cm 36. Find tree noncongruent polygons, ec wit n re of 4 squre units, on 6-y-6 geoord or 6-y-6 squre dot grid. 37. Lncelot wnts to mke pen for is pet, Isosceles. Wt is te re of te lrgest rectngulr pen tt Lncelot cn mke wit 100 meters of fencing if e uses strigt wll of te cstle for one side of te pen? 150 m 38. If you ve undred feet of rope to 39. Wic is etter fit (fills more of te ole): rrnge into te perimeter of eiter round peg in squre ole or squre peg in squre or circle, wic spe will give round ole? Eplin. you te mimum re? Eplin. 36. smple nswers: Ropes Circle. For te squre, 100 4s, s 5, A 5 65 ft. For te circle, 100 r, r 15.9, A (15.9) 794 ft. COMMON CORE STANDARDS G.GMD.1 G.C.5 A round peg in squre ole is etter fit. Te round peg fills out 78.5% of re of te squre ole, weres te squre peg fills only out 63.7% of te re of te round ole. 33. Are is 81 squre units. F (0, 0) y R (4, 15) O (4, 3) U (9, 5) Eercise 38 Some students my ve difficulty going from circumference to rdius. It migt e esier for tem to find te dimeter y dividing te circumference y nd ten lving to get te rdius. Eercise 39 Students my e tempted to try to solve tis prolem y sutrction, ut differences don t mke muc sense ecuse te sizes of te oles ren t given. Rtios re needed. If students re ving difficulty finding te re of te squre peg tt s in te round ole, suggest tt tey drw squre wit r s one side nd see ow muc of te originl squre would fit into tis new squre. Tey cn ten see tt te re of te squre peg is r.tey end up compring 4 wit. CHAPTER 8 REVIEW 473

56 Eercise 41 Tis eercise foresdows re rtios in Lesson A few students my e redy to see tt ecuse te rtio of liner dimensions is ,te 3 rtio ofres will e Eercises Encourge te use of dimensionl nlysis. Eercise 46 Students migt not recognize tt tey need to compre only te res of two circles. Tis eercise foresdows work wit proportions in Lesson A undle wit 1 te liner dimensions will ve 1 te re; it sould e 1 4 te price. 46. It s d del. r 1 44 cm. r cm, wic implies 4r 44 cm. Terefore r 1 r.te re ofte lrge undle is 4r cm.te comined re of two smll undles is r cm.tus e is getting lf s muc for te sme price. Eercise 47 Te oidizer cretes green ptin on te cones. Eercise 48 If students ren t seeing te dimensions of prticulr figure, suggest tt tey follow te projection lines from tt figure to one of te wlls were dimensions re given. Pinting will not tke plce on te prt of te wll of te lrger section tt is ginst te smller section. Tke Anoter Look EW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPTER 8 REVIEW CHAPT 40. Al Dente s Pizzeri sells pizz y te slice, ccording to te sign. Wic slice is te est del (te most pizz per dollr)? gint 41. If you need 8 oz of doug to mke 1-inc dimeter pizz, ow muc doug will you need to mke 16-inc pizz on crust of te sme tickness? out 14 oz 4. Wic is te iggest slice of pie: one-fourt of 6-inc dimeter pie, one-sit of n 8-inc dimeter pie, or one-eigt of 1-inc dimeter pie? Wic slice s te most crust long te curved edge? One-eigt of 1-inc dimeter pie; one-fourt of 6-inc pie nd one-eigt of 1-inc pie ot ve te sme lengt of crust, wic is longer tn one-sit of n 8-inc pie. 43. Te Hot-Air Blloon Clu t D Vinci Hig Scool s designed lloon for te nnul rce. Te pnels re regulr octgon, eigt squres, nd siteen isosceles trpezoids, nd clu memers will sew tem togeter to construct te lloon. Tey ve uilt scle model, s sown t rigt. Te pproimte dimensions of te four types of pnels re elow, sown in feet. 3. ft ft 1 ft ft 1 ft 0 ft 1 ft 5 ft 1 ft 1 ft. Wt will e te perimeter, to te nerest foot, of te lloon t its widest? Wt will e te perimeter, to te nerest foot, of te opening t te ottom of te lloon? 96 ft; 40 ft. Wt is te totl surfce re of te lloon to te nerest squre foot? 390 ft 44. You re producing 10,000 of tese metl wedges, nd you must electroplte tem wit tin lyer of ig-conducting silver. Te mesurements sown re in centimeters. Find te totl cost for silver, if silver plting costs $1 for ec 00 squre centimeters. Assume ec qudrilterl is rectngle. $ A possile pproc for te tringle: To mke se of constnt lengt, construct line j etween points A nd B. Mrk vector AB. Construct point C on line j. Trnslte point C to C y te mrked vector. Ten CC will e constnt. To crete constnt ltitude, construct point D not on line j. Construct line k troug point D prllel to line j. Construct point E on line k. Construct segments to form CCE of constnt se nd eigt. Ten ide points A, B, nd D nd lines j nd k. 1. For te prllelogrm, follow te sme steps s for te tringle, troug constructing point E. Ten trnslte E to E y te mrked vector. Connect points C, C, E, nd E to form prllelogrm CCEE. 474 CHAPTER 8 Are

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