51 CHAPTER 9 Geomety B C E A 5x x M x D F To contuct a golden ectangle, one in wic te atio of te lengt to te widt i equal to te atio of te lengt plu te widt to te lengt, egin wit a quae ABCD. Wit te point of te compae at M, te midpoint of AD, wing an ac of adiu MC to inteect te extenion of AD at F. Contuct a pependicula at F, and ave it inteect te extenion of BC at E. Ten ABEF i a golden ectangle wit atio 1 5. (See Section 5.4 fo moe on te golden atio.) To veify ti contuction, let AM x, o tat AD x. Ten, y te Pytagoean teoem, MC x x x 5x Since CF x 4x 5x. i an ac of te cicle wit adiu MC, MF MC 5x. Ten te atio of lengt AF to widt EF i AF x 5x EF x x x5 x x 1 5 x 1 5. 9.3 Peimete, Aea, and Cicumfeence Peimete and Aea In dealing wit plane figue, we ae often equied to find te ditance aound te figue, o te amount of uface coveed y te figue. Tee idea ae expeed a peimete and aea. Peimete and Aea Te peimete of a plane figue compoed of line egment i te um of te meaue of te line egment. Te aea of a plane figue i te meaue of te uface coveed y te figue. Peimete i meaued in linea unit, wile aea i meaued in quae unit. Te implet polygon i a tiangle. If a tiangle a ide of lengt a,, and c, ten to find it peimete we imply find te um of a,, and c. Ti can e expeed a a fomula, a own elow. Peimete of a Tiangle Te peimete P of a tiangle wit ide of lengt a,, and c i given y te fomula P a c. Since a ectangle i made up of two pai of ide wit eac ide in eac pai equal in lengt, te fomula fo te peimete of a ectangle may e tated a follow. Peimete of a Rectangle Te peimete P of a ectangle wit lengt and widt w i given y te fomula P w, o equivalently, P w. EXAMPLE 1 A plot of land i in te ape of a ectangle. If it a lengt 50 feet and widt 6 feet, ow muc fencing would e needed to completely encloe te plot? Since we mut find te ditance aound te plot of land, te fomula fo te peimete of a ectangle i needed. Sutitute 50 fo and 6 fo w to find P. P w P 50 6 P 100 5 P 15 50, w 6 Te peimete i 15 feet, o 15 feet of fencing i equied. a c P = a + + c P = + w o P = ( + w) w
9.3 Peimete, Aea, and Cicumfeence 513 A quae i a ectangle wit fou ide of equal lengt. Te fomula fo te peimete of a quae i a pecial cae of te fomula fo te peimete of a ectangle. Peimete of a Squae Te peimete P of a quae wit all ide of lengt i given y te fomula P 4. P = 4 EXAMPLE A quae a peimete 54 ince. Wat i te lengt of eac ide? Ue te fomula P 4 wit P 54, and olve fo. Eac ide a a meaue of 13.5 ince. Polem Solving P 4 54 4 13.5 P 54 Divide y 4. Te ix-tep metod of olving an applied polem uing algea can e ued to olve polem involving geometic figue. Fom: To: 1 + W FIGURE 9 W EXAMPLE 3 Te lengt of a ectangula-aped lael i 1 centimete moe tan twice te widt. Te peimete i 110 centimete. Find te lengt and te widt. Step 1: Read te polem. We mut find te lengt and te widt. Step : Aign a vaiale. Let W epeent te widt. Ten 1 W can epeent te lengt, ince te lengt i 1 centimete moe tan twice te widt. Figue 9 ow a diagam of te lael. Step 3: Wite an equation. In te fomula P w, eplace w wit W, wit 1 W, and P wit 110, ince te peimete i 110 centimete. 110 1 W W Step 4: Solve te equation. 110 4W W 110 6W 108 6W 18 W Ditiutive popety Comine tem. Sutact. Divide y 6. Step 5: State te anwe. Since W 18, te widt i 18 centimete and te lengt i 1 W 1 18 37 centimete. Step 6: Ceck. Becaue 37 i 1 moe tan twice 18, and ecaue te peimete 37 18 110 centimete, te anwe ae coect.
514 CHAPTER 9 Geomety 6 cm 4 cm 4 cm FIGURE 30 Defining te aea of a figue equie a aic unit of aea. One tat i commonly ued i te quae centimete, aeviated cm. One quae centimete, o 1 cm, i te aea of a quae one centimete on a ide. In place of 1cm, te aic unit of aea could ave een 1 in., 1 ft, 1 m, o any appopiate unit. A an example, we calculate te aea of te ectangle own in Figue 30. Uing te aic 1 cm unit, Figue 30 ow tat 4 quae, eac 1 cm on a ide, can e laid off oizontally wile 6 uc quae can e laid off vetically. A total of 4 4 6 of te mall quae ae needed to cove te lage ectangle. Tu, te aea of te lage ectangle i 4 cm. We can genealize te aove illutation to otain a fomula fo te aea of a ectangle. Aea of a Rectangle Te aea A of a ectangle wit lengt and widt w i given y te fomula A w. A = w w 3 7 1 15 15 10 10 FIGURE 31 40 Te fomula fo te aea of a ectangle A w can e ued to find fomula fo te aea of ote figue. Fo example, if te lette epeent te equal lengt of te ide of te quae, ten A. Aea of a Squae Te aea A of a quae wit all ide of lengt i given y te fomula A. A = EXAMPLE 4 Figue 31 ow te floo plan of a uilding, made up of vaiou ectangle. If eac lengt given i in mete, ow many quae mete of capet would e equied to capet te uilding? Te daed line in te figue eak up te floo aea into ectangle. Te aea of te vaiou ectangle tat eult ae 10 1 10 m, 3 10 30 m, 3 7 1 m, 15 5 375 m. Te total aea i 10 30 1 375 546 m, o 546 quae mete of capet would e needed. FIGURE 3 A mentioned ealie in ti capte, a paallelogam i a fou-ided figue aving ot pai of oppoite ide paallel. Since a paallelogam need not e a ectangle, te fomula fo te aea of a ectangle cannot e ued diectly fo a paallelogam. Howeve, ti fomula can e ued indiectly, a own in Figue 3. Cut off te tiangle in colo, and attac it at te igt. Te eulting figue i a ectangle aving te ame aea a te oiginal paallelogam.
9.3 Peimete, Aea, and Cicumfeence 515 Te eigt of te paallelogam i te pependicula ditance etween two of te paallel ide and i denoted y in te figue. Te widt of te ectangle equal te eigt of te paallelogam, and te lengt of te ectangle i te ae of te paallelogam, o A lengt widt ecome A ae eigt. Aea of a Paallelogam Te aea A of a paallelogam wit eigt and ae i given y te fomula A. A = (Note: i not te lengt of a ide.) 6 cm 15 cm FIGURE 33 B B EXAMPLE 5 Find te aea of te paallelogam in Figue 33. Hee te ae a a lengt of 15 centimete, wile te eigt i 6 centimete. Tu, 15 and 6. A 15 6 90 Te aea i 90 cm. Figue 34 ow ow we can find a fomula fo te aea of a tapezoid. Notice tat te figue a a wole i a paallelogam. It i made up of two tapezoid, eac of wic a eigt, ote ae, and longe ae B. Te aea of te paallelogam i found y multiplying te eigt y te ae of te paallelogam, B, tat i, B. Since te aea of te paallelogam i twice te aea of eac tapezoid, te aea of eac tapezoid i alf te aea of te paallelogam. FIGURE 34 3 cm 6 cm Aea of a Tapezoid Te aea A of a tapezoid wit paallel ae and B and eigt i given y te fomula A = 1_ ( + B) 9 cm A 1 B. B FIGURE 35 B A C FIGURE 36 EXAMPLE 6 Find te aea of te tapezoid in Figue 35. Hee 6, 3, and B 9. Tu, A 1 B 1 69 3 1 36. 61 Te aea of te tapezoid i 36 cm. Te fomula fo te aea of a tiangle can e found fom te fomula fo te aea of a paallelogam. In Figue 36 te tiangle woe vetice ae at A, B, and C a een oken into two pat, one own in colo and one own in gay. Repeating te pat own in colo and te pat in gay give a paallelogam. Te aea of ti paallelogam i A ae eigt, o A. Howeve, te paallelogam a twice te aea of te tiangle; in ote wod, te aea of te tiangle i alf te aea of te paallelogam.
516 CHAPTER 9 Geomety Aea of a Tiangle Te aea A of a tiangle wit eigt and ae i given y te fomula A = 1_ A 1. In eac cae, A 1. FIGURE 37 Wen applying te fomula fo te aea of a tiangle, ememe tat te eigt i te pependicula ditance etween a vetex and te oppoite ide (o te extenion of tat ide). See Figue 37. FOR FURTHER THOUGHT To ue te fomula fo te aea of a tiangle, A 1, we mut know te eigt of one of te ide of te tiangle. Suppoe tat we know only te lengt of te tee ide. I tee a way to detemine te aea fom only ti infomation? Te anwe i ye, and it lead u to te fomula known a Heon fomula. Heon of Alexandia lived duing te econd alf of te fit centuy A.D., and altoug te fomula i named afte im, tee i evidence tat it wa known to Acimede eveal centuie ealie. Heon Aea Fomula Let a,, and c e lengt of te ide of any tiangle. Let 1a c epeent te emipeimete. Ten te aea A of te tiangle i given y te fomula A ( a)( )( c). Fo example, if te ide of a tiangle meaue 6, 8, and 1 feet, we ave 16 8 1 13, and o te aea of te tiangle i A, wee A 1313 613 813 1 455 1.33 quae feet. Fo Goup Dicuion 1. If you cla i eld in a ectangula-aped oom, meaue te lengt and te widt, and ten multiply tem to find te aea. Now, meaue a diagonal of te oom, and ue Heon fomula wit te lengt, te widt, and te diagonal to find te aea of alf te oom. Doule ti eult. Do you aea calculation agee?. Fo a tiangle to exit, te um of te lengt of any two ide mut exceed te lengt of te emaining ide. Have alf of te cla ty to calculate te aea of a tiangle wit a 4, 8, and c 1, wile te ote alf tie to calculate te aea of a tiangle wit a 10, 0, and c 34. In ot cae, ue Heon fomula. Ten, dicu te eult, dawing diagam on te calkoad to uppot te eult otained. 3. Te Vietnam Vetean Memoial in Waington, D.C., i in te ape of an unencloed iocele tiangle. Te wall fom a V-ape, and eac wall meaue 46.75 feet. Te ditance etween te end of te wall i 438.14 feet. Ue Heon fomula to find te aea encloed y te tiangula ape. 46.75 ft 438.14 ft 46.75 ft
9.3 Peimete, Aea, and Cicumfeence 517 Cicumfeence of a Cicle Te ditance aound a cicle i called it cicumfeence (ate tan peimete ). To undetand te fomula fo te cicumfeence of a cicle, ue a piece of ting to meaue te ditance aound a cicle. Meaue it diamete and ten divide te cicumfeence y te diamete. Ti quotient i te ame, no matte wat te ize of te cicle. Te eult of ti meauement i an appoximation fo te nume. We ave cicumfeence C o altenatively, C d. diamete d, Since te diamete of a cicle meaue twice te adiu, we ave d. Tee elationip allow u to tate te following fomula fo te cicumfeence of a cicle. Cicumfeence of a Cicle Te cicumfeence C of a cicle of diamete d i given y te fomula C d. Alo, te cicumfeence C of a cicle of adiu i given y te fomula C. d C = πd C = π i not a ational nume. In ti capte we will ue 3.14 a an ap- wen one i equied. Recall tat poximation fo EXAMPLE 7 π (a) A cicle a diamete 1.6 cm. Find it cicumfeence. Ue. Since C d, C 3.141.6 39.564 o 39.6 centimete, ounded to te neaet tent. () Te adiu of a cicle i 1.7 m. Find it cicumfeence. Ue. C 3.141.7 10.7 Te cicumfeence i appoximately 10.7 mete. 3.14 3.14 (a) π A π () FIGURE 38 Aea of a Cicle Te fomula fo te aea of a cicle can e jutified a follow. Stat wit a cicle a own in Figue 38(a), divided into many equal pie-aped piece (ecto). Reaange te piece into an appoximate ectangle a own in Figue 38(). Te cicle a cicumfeence, o te lengt of te appoximate ectangle i one-alf of te cicumfeence, o 1, wile it widt i. Te aea of te appoximate ectangle i lengt time widt, o. By cooing malle and malle ecto, te figue ecome cloe and cloe to a ectangle, o it aea ecome cloe and cloe to. Ti limiting pocedue lead to te following fomula.
518 CHAPTER 9 Geomety Aea of a Cicle Te aea A of a cicle wit adiu i given y te fomula A. A. Polem Solving Te fomula fo te aea of a cicle can e ued to detemine te et value fo te money te next time you pucae a pizza. Te next example ue te idea of unit picing. EXAMPLE 8 Paw-Paw Jonny delive pizza. Te pice of an 8-inc diamete peppeoni pizza i $6.99, wile te pice of a 16-inc diamete pizza i $13.98. Wic i te ette uy? To detemine wic pizza i te ette value fo te money, we mut fit find te aea of eac, and divide te pice y te aea to detemine te pice pe quae inc. 8-inc diamete pizza aea 4 50.4 in. a Radiu i 18 4 ince. 16-inc diamete pizza aea 8 00.96 in. a Radiu i 116 8 ince. Te pice pe quae inc fo te 8-inc pizza i $6.9950.4 13.9, wile te pice pe quae inc fo te 16-inc pizza i $13.9800.96 7.0. Teefoe, te 16-inc pizza i te ette uy, ince it cot appoximately alf a muc pe quae inc.