00020002010010Max_Triangle.ACT0001020010Max_Triangle.EAC0100000017e9 Max_Triangle.EACACT(+ 8< $]I ('.[i&,Paper folding and maximum area of a t4\< AuthorS<"A Problem suggested by  )Anothny HarradineB"JSchool of Mathematics@ Prince AlfrcCollegD, f Australia^he-Activity prepaO Wei-Chi Yang"URL: http://www. ford.edu/wy*f e-mail: @'C\ Sound TrackNRX\"Professor Jonathan Lewin ! Kennesaw State University (USA) H e-mail: l=s@mindspring.comMURL: http://www.mq-movies'[ Objectus[$M1. We will use ClassPad to demonstrhow dynamic geometry feature chelp -students formulways of solving a problem.<M2agqapproach (with no Calculus)Te thisU[ A Question:JE We fold a piece of rectangular paper (open the geometry below to seeUsample) from up?left-hand cornP7Mbasq i. SlidP+along \;5. FiNgest trile DEFat canrmed.\1Tap oisstrip play animationΈ #A `WW RbVPQ "PPtI0D/vKr  A 'L'cC ux y " +! 4(=(#``g`g0qQH1C001YwP<FP0Vg`ae$wEle&`U"HDW`4Tl$9ipYQxYt@$@6Twsd@e1TiWg7P 9$BG 5sbr(q(H!fqD9y`D0"PP7`v@ VRv@ $p70dPH`<Y<S"Dq0GQA$(  c P  r8%3I0 @pwR0p$bHT p`eȆ<v%9  VH4I=`g l#ԆxCg$Qh#$3Xy 7̆u#p"Txe8x`(!T$B Gq@u3$bY%3I0) $&P aAp$ 5aw $8; '@$ 0uhE<aA`HDT3P` #)y rv9  4b6(tQO`b`QChlF Coord: bH jFj$9ie5j( YK@ ]H"YD#dBY)$`aY   `)`DA $8w !H  *A@168+0&`IS7` G@ GY " !@#* !<a  Y(0W 3H!C%pU577ʼnňYH0` ӗCa h$@x W2gf@  93@%6,d%&`"@3y4GO EFD:   j jiqifw ve! ysCiAu E&" `gSxG@7j0  #`)a) Y(0V N 90S9  2@ 6'bTSW `Vg)8 B Area DEF: e }@m 2ʐ*"*2pĘh6   Y`)`5( Y$8w!` z""#H qE'rp!c(drqRS|$ ! @6qi&cE% uR0 Y&wTg.A #%N@ J`""&x(@t'" Fz Y"gx bh9@V 0 YV`q _g ll`)a H<(0WHN((ʙN$>(#) @  '*H D@269#d `)$`aPY@G+YȠY s&m,.)H.6-ĘH.4 Ȑ `)`DA $8w. J/``z."0;5@#*1Q !"Y`Јt2`3 1 aSaS  1( Q( R;Y4DH! M5Z"@i6puCyTdWV#W YdG.36 B`)`$ Y IC.`7H GsY+Y;XGU}8@Y9o:ve!gqifxr21 c ;`< @  :=75> Y46T(<;l986?t@ 6`)`G Y ID.`AF!BG%;P|B1H!CKoň@!<E&gSxF " @6qifw% ve! ysCiA.A GNH!J`Hhǘ0I  `)`V40 $8w lJ@J`K0LEMH Ei"8FD$6TBzl&NOP"6 `)`V40 $8w#' Q4H! PO".Rb@#.AtBR\S@!&nART1&Tz!@ zU 8@Ϝx %f!$YV֘\`W$Ƙ0= W2gXU'Ÿ'1'3 2.WVUCB@  A$ -5!1PFIE76:5.'*#$  (0 [ Open the geometry strip above:& 1. Notice+at AE=EF and AG=GF.M>2. We would like to fi'alargest area ofutriangle EFD.[Hint: P Reflect point E alongline segment DCconsidervpossible area of EFE'.[ Exploration:L-Play the anim . (Tap ongeometry strip above)=>View=>A4 UI.)T<-NoticeVat[_D varies when  changes. DU-Can you make conjecture5willV trianglecachieveitsDximum?]solvis problem by using onlyargumentDQ==== 1[(VWe demonstrate this by collecting e x-coordin$of point F and$areaEFD _afterC animation. A2!dragCdrop ,sidirly3back to geometry ip.[E-Cuctlinat perpendiculao DCpasses rough F.[S-Playچpep>( UI)'noticuWmaximum occurs#whenVgleLabout 30grees[ Solution:H-We construct the reflec% E' of E alongline segment DC (or D'C).P>PnoticeMaR triangle EFRis an oceles sin, EF=E'F.[GVobservH maximum areaFD occurs whenh equilateral. 'Or0M 30 degreesӑD Remarks: U PConsider, f-given base EE'. T th{;'largest possible for base EE', so it mu"?be isosceles, and[OEF=FE'. But if you consider EF as theU, n[ = 4So# t. V-There5 triangEl is equilateral. [  2;eActfH<,a,b,c,x(a^(2)+x^(2))+((b-x)^(2)+c^(2))010008main.ACT0001020012eActivity Save.EAC0100000017e9 Max_Triangle.EACACT(+ 8< $]I ('.[i&,Paper folding and maximum area of a t4\< AuthorS<"A Problem suggested by  )Anothny HarradineB"JSchool of Mathematics@ Prince AlfrcCollegD, f Australia^he-Activity prepaO Wei-Chi Yang"URL: http://www. ford.edu/wy*f e-mail: @'C\ Sound TrackNRX\"Professor Jonathan Lewin ! Kennesaw State University (USA) H e-mail: l=s@mindspring.comMURL: http://www.mq-movies'[ Objectus[$M1. We will use ClassPad to demonstrhow dynamic geometry feature chelp -students formulways of solving a problem.<M2agqapproach (with no Calculus)Te thisU[ A Question:JE We fold a piece of rectangular paper (open the geometry below to seeUsample) from up?left-hand cornP7Mbasq i. SlidP+along \;5. FiNgest trile DEFat canrmed.\1Tap oisstrip play animationΈ #A `WW RbVPQ "PPtI0D/vKr  A 'L'cC ux y " +! 4(=(#``g`g0qQH1C001YwP<FP0Vg`ae$wEle&`U"HDW`4Tl$9ipYQxYt@$@6Twsd@e1TiWg7P 9$BG 5sbr(q(H!fqD9y`D0"PP7`v@ VRv@ $p70dPH`<Y<S"Dq0GQA$(  c P  r8%3I0 @pwR0p$bHT p`eȆ<v%9  VH4I=`g l#ԆxCg$Qh#$3Xy 7̆u#p"Txe8x`(!T$B Gq@u3$bY%3I0) $&P aAp$ 5aw $8; '@$ 0uhE<aA`HDT3P` #)y rv9  4b6(tQO`b`QChlF Coord: bH jFj$9ie5j( YK@ ]H"YD#dBY)$`aY   `)`DA $8w !H  *A@168+0&`IS7` G@ GY " !@#* !<a  Y(0W 3H!C%pU577ʼnňYH0` ӗCa h$@x W2gf@  93@%6,d%&`"@3y4GO EFD:   j jiqifw ve! ysCiAu E&" `gSxG@7j0  #`)a) Y(0V N 90S9  2@ 6'bTSW `Vg)8 B Area DEF: e }@m 2ʐ*"*2pĘh6   Y`)`5( Y$8w!` z""#H qE'rp!c(drqRS|$ ! @6qi&cE% uR0 Y&wTg.A #%N@ J`""&x(@t'" Fz Y"gx bh9@V 0 YV`q _g ll`)a H<(0WHN((ʙN$>(#) @  '*H D@269#d `)$`aPY@G+YȠY s&m,.)H.6-ĘH.4 Ȑ `)`DA $8w. J/``z."0;5@#*1Q !"Y`Јt2`3 1 aSaS  1( Q( R;Y4DH! M5Z"@i6puCyTdWV#W YdG.36 B`)`$ Y IC.`7H GsY+Y;XGU}8@Y9o:ve!gqifxr21 c ;`< @  :=75> Y46T(<;l986?t@ 6`)`G Y ID.`AF!BG%;P|B1H!CKoň@!<E&gSxF " @6qifw% ve! ysCiA.A GNH!J`Hhǘ0I  `)`V40 $8w lJ@J`K0LEMH Ei"8FD$6TBzl&NOP"6 `)`V40 $8w#' Q4H! PO".Rb@#.AtBR\S@!&nART1&Tz!@ zU 8@Ϝx %f!$YV֘\`W$Ƙ0= W2gXU'Ÿ'1'3 2.WVUCB@  A$ -5!1PFIE76:5.'*#$  (0 [ Open the geometry strip above:& 1. Notice+at AE=EF and AG=GF.M>2. We would like to fi'alargest area ofutriangle EFD.[Hint: P Reflect point E alongline segment DCconsidervpossible area of EFE'.[ Exploration:L-Play the anim . (Tap ongeometry strip above)=>View=>A4 UI.)T<-NoticeVat[_D varies when  changes. DU-Can you make conjecture5willV trianglecachieveitsDximum?]solvis problem by using onlyargumentDQ==== 1[(VWe demonstrate this by collecting e x-coordin$of point F and$areaEFD _afterC animation. A2!dragCdrop ,sidirly3back to geometry ip.[E-Cuctlinat perpendiculao DCpasses rough F.[S-Playچpep>( UI)'noticuWmaximum occurs#whenVgleLabout 30grees[ Solution:H-We construct the reflec% E' of E alongline segment DC (or D'C).P>PnoticeMaR triangle EFRis an oceles sin, EF=E'F.[GVobservH maximum areaFD occurs whenh equilateral. 'Or0M 30 degreesӑD Remarks: U PConsider, f-given base EE'. T th{;'largest possible for base EE', so it mu"?be isosceles, and[OEF=FE'. But if you consider EF as theU, n[ = 4So# t. V-There5 triangEl is equilateral. 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