{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 50 "High School Modul es > Geometry by Gregory A. Moore" }{TEXT 265 1 "\n" }}{PARA 3 "" 0 " " {TEXT -1 4 " " }{TEXT 256 16 "Axes of Symmetry" }}{PARA 0 "" 0 " " {TEXT -1 99 "\nAn exploration of the axes of symmetry of regular pol ygons, and discussion of the number of axes.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section first. \+ Although there will be no output immediately, these definitions are us ed later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart; with(plots): with(geometry ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "sbl := 'axes = none , scaling = constrained, \n thickness = 3, color = COLOR(RGB, \+ .5,.4,.8)':\nsgy := 'axes = none, scaling = constrained, \n th ickness = 3, color = COLOR(RGB, .5,.5,.6)':\nsbe := 'axes = none, sc aling = constrained, \n thickness = 3, color = COLOR(RGB, .8,.7, .5)':\nsor := 'axes = none, scaling = constrained, \n thicknes s = 3, color = COLOR(RGB, .9,.7,.1)':\nsgn := 'axes = none, scaling \+ = constrained, \n thickness = 3, color = COLOR(RGB, .4,.7,.4)': " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "cbl := 'color = COLOR( RGB, .5,.4,.8)':\ncgy := 'color = COLOR(RGB, .5,.5,.6)':\ncbe := 'colo r = COLOR(RGB, .8,.7,.5)':\ncor := 'color = COLOR(RGB, .9,.7,.1)':\ncy l := 'color = COLOR(RGB, .9,.8,.1)':\ncgn := 'color = COLOR(RGB, .5,.8 ,.5)':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "scanft := 'scali ng = constrained, axes = none':\nscanft := 'filled = true, axes = none , scaling = constrained':\nscant2 := 'thickness = 2, axes = none, scal ing = constrained':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "point(o,0,0);\ncircle( UC ,[ o,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "RegPolyPlot := \+ proc( N )\n local k, n, ngon, c;\n ngon := n -> [seq([ cos(2*Pi*i/n ), sin(2*Pi*i/n) ], i = 1..n)]: \n c := n -> COLOR(RGB, .2+n/(2*N), \+ .2+n/(2*N), .5);\n display([ seq( polygonplot(ngon(n), color=c(n)), n = 3..N)],\n scaling = constrained, axes = none);\nend proc: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ngon := n -> [seq([ c os(2*Pi*i/n), sin(2*Pi*i/n) ], i = 1..n)]:\n " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 508 "PolyVAPlot := proc( N )\n local c1, k, Po ly, theta, theta2,scl;\n c1 := n -> COLOR(RGB, .2+n/(1*N), 0, .2+n/(1 *N) );\n Poly := ngon(N): scl := 1.2;\n for k from 1 to N do\n \+ theta := 2*Pi*k/N; theta2 := theta + Pi;\n VA||k := plot( [[cos (theta),sin(theta)],[scl*cos(theta2),scl*sin(theta2)]],\n \+ color=c1(k));\n od;\n display([ polygonplot(Poly, color=COLOR(RGB,.79,.76,.65 )),\n seq( VA||k, k=1..N)\n \+ ],scaling = constrained, axes = none);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 739 "PolySAPlot := proc( N )\n local c2,k,Poly,SA,theta,thetaB,theta2,theta2B,scl,x1,x2,y1,y2;\n c2 := n -> COLOR(RGB, .2+n/(2*N), .1+n/(2*N), .3 );\n Poly := ngon(N): \+ scl := 1.2;\n for k from 1 to N/2 do\n theta := 2*Pi*k/N; \+ theta2 := theta + Pi;\n thetaB := 2*Pi*(k+1)/N; theta2B := the taB + Pi;\n x1 := scl*(cos(theta) + cos(thetaB) )/2;\n x2 := \+ scl*(cos(theta2) + cos(theta2B) )/2;\n y1 := scl*(sin(theta) + s in(thetaB) )/2;\n y2 := scl*(sin(theta2) + sin(theta2B) )/2;\n\n \+ SA||k := plot( [[x1,y1],[x2,y2]], color=c2(k), linestyle = 3);\n \+ od;\n display([ polygonplot(Poly,color=COLOR(RGB,.79,.76,.61 )),\n \+ seq( SA||k, k=1..N/2)\n ],scaling = constrained, axe s = none);\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1238 "PolyAxesPlot := pro c( N )\n local c1,c2, k, Poly,VA,SA, scl,\n theta,thetaB,the ta2,theta2B,x1,x2,y1,y2;\n c1 := n -> COLOR(RGB, .2+n/(1*N), 0, .2+n/ (1*N) );\n c2 := n -> COLOR(RGB, .2+n/(2*N), .1+n/(2*N), .3 );\n Pol y := ngon(N): scl := 1.2;\n for k from 1 to N do\n theta := 2 *Pi*k/N; theta2 := theta + Pi;\n VA||k := plot( \n \+ [[cos(theta),sin(theta)],[scl*cos(theta2),scl*sin(theta2)]],\n \+ color=c1(k));\n od;\n if( (N mod 2) = 0) then\n for k \+ from 1 to N/2 do\n theta := 2*Pi*k/N; theta2 := theta + \+ Pi;\n thetaB := 2*Pi*(k+1)/N; theta2B := thetaB + Pi;\n x 1 := scl*(cos(theta) + cos(thetaB) )/2;\n x2 := scl*(cos(thet a2) + cos(theta2B) )/2;\n y1 := scl*(sin(theta) + sin(thetaB) \+ )/2;\n y2 := scl*(sin(theta2) + sin(theta2B) )/2;\n SA||k : = plot( [[x1,y1],[x2,y2]],color=c2(k), linestyle = 3);\n od;\n di splay([ polygonplot(Poly,color=COLOR(RGB,.79,.76,.65 )),\n \+ seq( VA||k, k=1..N), seq( SA||k, k=1..N/2)\n ],scaling = cons trained, axes = none);\n else\n display([ polygonplot(Poly,color=C OLOR(RGB,.79,.76,.65 )),\n seq( VA||k, k=1..N)\n ] ,scaling = constrained, axes = none);\n fi;\n \nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 718 "PolyMiniPlot := proc( N )\n local c2,k,Poly,SA,t heta,thetaB,theta2,theta2B,scl,x1,x2,y1,y2;\n c2 := n -> COLOR(RGB, . 2+n/(2*N), .3+n/(2*N),0 );\n Poly := ngon(N): scl := 1.2;\n for k from 1 to N do\n theta := 2*Pi*k/N; theta2 := theta + Pi ;\n thetaB := 2*Pi*(k+1)/N; theta2B := theta + Pi;\n x1 := (co s(theta) + cos(thetaB) )/2;\n x2 := (cos(theta2) + cos(theta2B) ) /2;\n y1 := (sin(theta) + sin(thetaB) )/2;\n y2 := (sin(theta2 ) + sin(theta2B) )/2;\n\n SA||k := plot( [[x1,y1],[x2/2,y2/2]], co lor=c2(k));\n od;\n display([ polygonplot(Poly,color=COLOR(RGB,.79,. 76,.63 )),\n seq( SA||k, k=1..N)\n ],scaling = con strained, axes = none);\nend proc:\n\n\nPolyMiniPlot(6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 28 "1. Regular Polygons (review)" }}{PARA 0 "" 0 "" {TEXT -1 150 "\nRegular polygons are polygons where all the si des are equal and all the interior angles are equal. A five sided regu lar polygon is called a pentagon." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "RegularPolygon( RegPoly,5,o,1):\ndisplay( draw(RegPol y, scanft, cbl ));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 207 "\nIf we draw a number of regular polygons on the same \+ diagram we can see the progression. Here are regular polygons with 3 t o 10 sides, and then 3 to 30 sides. They seem to fill the unit circle \+ more and more." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "RegPolyPlo t(6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "RegPolyPlot(10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "RegPolyPlot(25);" }}} {PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 264 39 "2. Axes of Symmetry : Through Vertices" }}{PARA 0 "" 0 "" {TEXT -1 353 "\nWith a regular polygon, you can choose any v ertex, then draw a line through that vertex and the center of the poly gon (which is the same as the center of the circle that would circumsc ribe it), then continue the line through to the other side of the poly gon. \n\nIf the number of sides is an odd number, then the opposite si de will pass through a side. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "PolyVAPlot(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P olyVAPlot(7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "PolyVAPlot (13);" }}}{PARA 0 "" 0 "" {TEXT -1 200 "\nSo odd regular polys have on e axis of symmetry for eac vertex. A 129-sided polygon will have 129 a xes of symmetry of this type.\n\nIf the number of sides is even, it wi ll pass through another vertex. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "PolyVAPlot(8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "PolyVAPlot(12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " PolyVAPlot(10);" }}}{PARA 0 "" 0 "" {TEXT -1 141 "\n This means that t here are half as many axes of this type. For example a polygon with 12 sides will have 6 axes which pass through vertices." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 263 36 "3. Axes of Symmetry : T hrough Sides" }}{PARA 0 "" 0 "" {TEXT -1 172 "\nEven the number of sid es is even, there are also axes of symmetry that pass through the midp oints of sides. This is not true for regular polys with an odd numbere d sides.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "PolySAPlot(6); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "PolySAPlot(8);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "PolySAPlot(12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "PolySAPlot(20);" }}}{PARA 0 "" 0 " " {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 43 "4. Axes of Symmetry : All Axes of Symmetry" }}{PARA 0 "" 0 " " {TEXT -1 209 "\nAs we saw above there is a difference for a regular \+ n-sided polygon depending on whether n is even or odd. If n is odd, th ere are n axes of symmetry - each of which passes through a vertex and opposite side.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "PolyAxe sPlot(7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "PolyAxesPlot(1 1);" }}}{PARA 0 "" 0 "" {TEXT -1 186 "\n\nOn the other hand, if n is e ven, there are two types of axes of symmetry - the the n/2 that pass t hrough two vertices, and the n/2 which pass through two opposite sides - a total of n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "PolyAxes Plot(4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "PolyAxesPlot(6) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "PolyAxesPlot(8);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "PolyAxesPlot(12);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "PolyAxesPlot(20);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 2002 Water loo Maple Inc " }}}{MARK "0 1" 22 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }