{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 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 4 "" 0 "" {TEXT -1 19 "Module 2 : Geometry" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 21 "202 : Historic Que sts" }}{PARA 0 "" 0 "" {TEXT -1 22 " 2000 Gregory A. Moore" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________ __________________________________________________________________" }} {PARA 4 "" 0 "" {TEXT -1 31 "A. Proof of Pythagoras' Theorem" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "In thi s section we will create a famous diagram which can be an aid in provi ng the validity of the theorem of Pythagoras. This is one of more than 100 proofs of this famous and fundamental theorem." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a := 5; b := 7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "with(plots):\ndisplay( polygonplot( [[0,0],[b,0],[0,a]],color = red ),\n polygonplot( [[b,0],[a+b,0],[a+b,b]], color = green ), \n polygonplot( [[a+b,b],[a+b,a+b],[a,a+b]], color = blue ),\n polygon plot( [[0,a],[a,a+b],[0,a+b]], color = violet ),\n polygonplot(\{[[b, 0],[a+b,b],[a,a+b],[0,a]]\},color = yellow), scaling= constrained);" } }{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+ redefined\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(- %)POLYGONSG6$7%7$$\"\"!F)F(7$$\"\"(F)F(7$F($\"\"&F)-%'COLOURG6&%$RGBG$ \"*++++\"!\")F(F(-F$6$7%F*7$$\"#7F)F(7$F;F+-F16&F3F(F4F(-F$6$7%F=7$F;F ;7$F.F;-F16&F3F(F(F4-F$6$7%F-FD7$F(F;-F16&F3$\")#R!)4$F6$\")t8V=F6FM-F $6$7&F*F=FDF--F16&F3F4F4F(-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________ _______________________________________________________________" }} {PARA 4 "" 0 "" {TEXT -1 34 "B. Area of A Circle Using Polygons" }} {PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________ __________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 627 "Its easy enough to compute the area of triangles and rectangle s. In fact, the area of any region bounded by straight lines can be co mputed by cutting it into triangles or other simple shapes. However, a curved shape like a circle defies this method. A crude estimate for t he area of circle would be to find the area of an inscribed regular po lygon - that is, a polygon which has all of its vertices on the circle , and each edge is same length. We can use Maple\325s built-in command to generate regular polygons. Then if we increase the number of sides , the area of the polygon becomes closer and closer to the area of the circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "In the geometry package, the name for a geometric object is usual ly the first name in the parameter list. Also the geometry package use s draw rather than plot or polygonplot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "with(geometry): point( orig,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%origG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RegularPolygon( P, 12, orig, 1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"PG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "draw( P );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "61-%'CURVESG6%7$7$$\"\"\"\"\"!$F*F*7$$\"+SSDg')!#5$\"++++ +]F/-%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-F$6%7$F,7$F 0F-F2F6-F$6%7$F@7$F+F(F2F6-F$6%7$FD7$$!+++++]F/F-F2F6-F$6%7$FH7$$!+SSD g')F/F0F2F6-F$6%7$FN7$$!\"\"F*F+F2F6-F$6%7$FT7$FOFIF2F6-F$6%7$FZ7$FIFO F2F6-F$6%7$Fhn7$F+FUF2F6-F$6%7$F\\o7$F0FOF2F6-F$6%7$F`o7$F-FIF2F6-F$6% 7$FdoF'F2F6-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$; FUF(Fcp" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curv e 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Every regular polygon wi ll fit inside of a circle and can be considered to be inscribed in the circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(geometry): point(orig, 0, 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%origG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "circle( C, [point( origin, 0, 0), 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"CG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "## # WARNING: calls to `C` for generating C code should be replaced by co degen[C]\ndraw( \{C(color = black),P(color = gold, filled = true)\} ); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6% 7U7$$\"\"\"\"\"!$F*F*7$$\"+8q9@**!#5$\"+OBL`7F/7$$\"+6;$eo*F/$\"+t))*o [#F/7$$\"+f[w(H*F/$\"+FbC\"o$F/7$$\"++o1j()F/$\"+Tn`<[F/7$$\"+V*p,4)F/ $\"+CD&y(eF/7$$\"+tio*G(F/$\"+g5ZXoF/7$$\"+'*)RUP'F/$\"+HC80xF/7$$\"+^ zEe`F/$\"+a#zKW)F/7$$\"+=HzdUF/$\"+B0F[!*F/7$$\"+Q*p,4$F/$\"+l^c5&*F/7 $$\"+UJ\"Q(=F/$\"+3D(G#)*F/7$$\"+D>0zi!#6$\"+%Gn-)**F/7$$!+m>0ziFaoFbo 7$$!+YJ\"Q(=F/$\"+2D(G#)*F/7$$!+U*p,4$F/$\"+j^c5&*F/7$$!+8HzdUF/$\"+E0 F[!*F/7$$!+YzEe`F/$\"+d#zKW)F/7$$!+$*)RUP'F/$\"+JC80xF/7$$!+$G'o*G(F/$ \"+]5ZXoF/7$$!+]*p,4)F/$\"+9D&y(eF/7$$!+0o1j()F/$\"+Kn`<[F/7$$!+i[w(H* F/$\"+>bC\"o$F/7$$!+8;$eo*F/$\"+l))*o[#F/7$$!+9q9@**F/$\"+IBL`7F/7$$! \"\"F*$!+:w1-T!#>7$$!+8q9@**F/$!+QBL`7F/7$$!+6;$eo*F/$!+t))*o[#F/7$$!+ f[w(H*F/$!+FbC\"o$F/7$$!+,o1j()F/$!+Sn`<[F/7$$!+Y*p,4)F/$!+?D&y(eF/7$$ !+xio*G(F/$!+c5ZXoF/7$$!+())RUP'F/$!+PC80xF/7$$!+RzEe`F/$!+i#zKW)F/7$$ !+1HzdUF/$!+H0F[!*F/7$$!+M*p,4$F/$!+m^c5&*F/7$$!+QJ\"Q(=F/$!+4D(G#)*F/ 7$$!+%)=0ziFao$!+&Gn-)**F/7$$\"+2?0ziFao$!+%Gn-)**F/7$$\"+]J\"Q(=F/$!+ 2D(G#)*F/7$$\"+Y*p,4$F/$!+i^c5&*F/7$$\"+Fc]l-%(SCALINGG6#%,CONSTRAINED G-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F_sF(Fc^l" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Lets look at several r elated regular polygons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 89 "Notice that for the 20 sided polygon, the area is \+ 99% the same as the area of the circle!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(geometry): point( orig, 0, 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%origG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "RegularPolygon(Poly5,5,point(Origin ,0,0),1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Poly5G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "RegularPolygon(Poly10,10,point(Orig in,0,0),1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'Poly10G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "RegularPolygon(Poly20,20,point(Orig in,0,0),1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'Poly20G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "### WARNING: calls to `C` for gene rating C code should be replaced by codegen[C]\ndraw( \{C(color = blac k),Poly5(color= blue),\n Poly10(color =green),Poly20(color= red)\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6I-%'CURVESG6%7U 7$$\"\"\"\"\"!$F*F*7$$\"+8q9@**!#5$\"+OBL`7F/7$$\"+6;$eo*F/$\"+t))*o[# F/7$$\"+f[w(H*F/$\"+FbC\"o$F/7$$\"++o1j()F/$\"+Tn`<[F/7$$\"+V*p,4)F/$ \"+CD&y(eF/7$$\"+tio*G(F/$\"+g5ZXoF/7$$\"+'*)RUP'F/$\"+HC80xF/7$$\"+^z Ee`F/$\"+a#zKW)F/7$$\"+=HzdUF/$\"+B0F[!*F/7$$\"+Q*p,4$F/$\"+l^c5&*F/7$ $\"+UJ\"Q(=F/$\"+3D(G#)*F/7$$\"+D>0zi!#6$\"+%Gn-)**F/7$$!+m>0ziFaoFbo7 $$!+YJ\"Q(=F/$\"+2D(G#)*F/7$$!+U*p,4$F/$\"+j^c5&*F/7$$!+8HzdUF/$\"+E0F [!*F/7$$!+YzEe`F/$\"+d#zKW)F/7$$!+$*)RUP'F/$\"+JC80xF/7$$!+$G'o*G(F/$ \"+]5ZXoF/7$$!+]*p,4)F/$\"+9D&y(eF/7$$!+0o1j()F/$\"+Kn`<[F/7$$!+i[w(H* F/$\"+>bC\"o$F/7$$!+8;$eo*F/$\"+l))*o[#F/7$$!+9q9@**F/$\"+IBL`7F/7$$! \"\"F*$!+:w1-T!#>7$$!+8q9@**F/$!+QBL`7F/7$$!+6;$eo*F/$!+t))*o[#F/7$$!+ f[w(H*F/$!+FbC\"o$F/7$$!+,o1j()F/$!+Sn`<[F/7$$!+Y*p,4)F/$!+?D&y(eF/7$$ !+xio*G(F/$!+c5ZXoF/7$$!+())RUP'F/$!+PC80xF/7$$!+RzEe`F/$!+i#zKW)F/7$$ !+1HzdUF/$!+H0F[!*F/7$$!+M*p,4$F/$!+m^c5&*F/7$$!+QJ\"Q(=F/$!+4D(G#)*F/ 7$$!+%)=0ziFao$!+&Gn-)**F/7$$\"+2?0ziFao$!+%Gn-)**F/7$$\"+]J\"Q(=F/$!+ 2D(G#)*F/7$$\"+Y*p,4$F/$!+i^c5&*F/7$$\"+ " 0 "" {MPLTEXT 1 0 66 " evalf( area(P oly5)); evalf( area(Poly10)); evalf( area(Poly20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"HTwP#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+ii#*QH!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W*p,4$!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Notice th at these numbers are less than but approaching ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 224 "In the diagram above, yo u might have noticed that the greater the number of sides in the regul ar polygon, the closer it resembles the circle its inscribed in. We ca n compare the area of the polygons to the area of the circle" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " e valf( area(Poly5)) / evalf(area(C)); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'Gn#ov!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " eva lf( area(Poly10)) / evalf(area(C)); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+QG*[N*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " eva lf( area(Poly20)) / evalf(area(C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+IkJO)*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "We can do the same thing with perimeters. The perimeters of the polygons should be getting close to 2 (the circumference of th e unit circle)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 81 " evalf( perimeter(Poly5)); evalf( perimeter(Po ly10)); evalf( perimeter(Poly20));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+CD&y(e!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)))R.='!\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/'ytD'!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 " evalf(perimeter(Poly5)/(2*Pi)); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SG*[N*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " evalf(perimeter(Poly10)/(2*Pi)); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IkJO)*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " evalf(perimeter(Poly20)/(2*Pi));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+at#*e**!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 27 "C . Impossible Constructions" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________ ______________________________________________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1170 "Its pos sible to construct many geometric figures using only an unmarked strai ght edge and a compass . From antiquity there were three constructions which were sought after but never found :\n\011 trisecting an angle\n \011 squaring a circle\n\011 doubling a cube\nIn the late 1800\325s, i t was proven that all three of these are impossible to construct with \+ straight edge and compass using the methods of abstract algebra. When \+ we say something is impossible in mathematics, this is different than \+ something being impossible in science. For example, its impossible to \+ turn lead into gold - at least with our current technology. Howvever, \+ many things we take for granted today such as travelling faster than t he speed of sound, viewing a scene as is happens on the other side of \+ the world, etc. were once impossible\325 from a scientific point of vi ew. In mathematics, something being impossible is a much stronger cond ition because it means not just that it can\325t be done now, but also that it will never be done in the future. The three construction prob lems listed above are indeed impossible - although that does not preve nt various (uninformed) people from attempting to find solutions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 243 "Because \+ these constructions are impossible, we will not be able to solve them \+ using the straightedge and compass. However, using Maple we can approx imate to a high degree of accuracy the solutioins to better understand the unsuccessful quests." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 16 "TRISECT AN ANGLE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "To trisect an arbitrary angle mean s to divide it evenly into three sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 224 "We define two points which wil l determine an angle from the origin.\nThen we compute the angle betwe en the lines, and construct the two angle trisectors. You can change t he initial points A and B to create angles to trisect\325!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " res tart; with(geometry):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " p oint( orig, 0, 0); point( A, 5, 1); point( B, 2, 7); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%origG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"BG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 81 " line(L1,[orig,A]); line(L2,[orig,B]); \n line(x_a xis, [orig, point(xx, 10, 0)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#L 1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#L2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'x_axisG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " a2 := evalf( FindAngle(L2, x_axis)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " a1 := evalf( FindAngle(L1, x_axis)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " angle_rad := evalf( a2 - a1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*angle_radG$\"+365&4\"!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " draw( [L1(color=green),L2(c olor=red)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6' -%'CURVESG6%7U7$$\"#5\"\"!$\"+++++?!\"*7$$\"+++++'*F-$\"++++?>F-7$$\"+ ++++#*F-$\"++++S=F-7$$\"+++++))F-$\"++++gF-7$$FhoF*$!+++++?F--%&STYLEG6 #%%LINEG-%'COLOURG6&%$RGBGFar$\"*++++\"FgwFar-F$6%7U7$F($\"+++++NFgw7$ F/$\"++++gLFgw7$F4$\"++++?KFgw7$F9$\"++++!3$Fgw7$F>$\"++++SHFgw7$FC$F \\qFgw7$FH$\"++++gEFgw7$FM$\"++++?DFgw7$FR$\"++++!Q#Fgw7$FW$\"++++SAFg w7$Ffn$\"+++++@Fgw7$F[o$\"++++g>Fgw7$F`o$\"++++?=Fgw7$Feo$FAFgw7$Fjo$ \"++++S:Fgw7$F^p$\"+++++9Fgw7$Fcp$\"++++g7Fgw7$Fgp$F^oFgw7$F[q$\"+++++ )*F-7$F_qF>7$Fcq$\"+++++qF-7$FEF[o7$Fhn$\"+++++UF-7$F`pF[q7$Feq$Fh\\lF -F`r7$Fer$!+++++9F-7$FjrF^t7$F^s$!+++++UF-7$FcsF]v7$Fgs$!+++++qF-7$Fjs Fiw7$F^t$!+++++)*F-7$Fct$F_vFgw7$Fgt$!++++g7Fgw7$F\\u$F^^lFgw7$F_u$!++ ++S:Fgw7$Fdu$F\\xFgw7$Fhu$!++++?=Fgw7$F]v$!++++g>Fgw7$Fav$!+++++@Fgw7$ Fdv$!++++SAFgw7$Fhv$!++++!Q#Fgw7$F]w$!++++?DFgw7$Faw$!++++gEFgw7$Ffw$F _tFgw7$Fiw$!++++SHFgw7$F^x$!++++!3$Fgw7$Fbx$!++++?KFgw7$Fgx$!++++gLFgw 7$F[y$!+++++NFgwF^y-Fcy6&FeyFfyFarFar-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6 #%,CONSTRAINEDG-%%VIEWG6$;F[yF(Fjbl" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " theta := angle_rad/3:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 " point( C, cos( a1 + theta), sin(a1 + theta)):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " point( D, cos( a1 + 2*the ta), sin(a1 + 2*theta)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " line( s1, [orig, C]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " line( s2, [orig, D]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " \+ draw( [L1,L2, s1, s2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6%7U7$$\"#5\"\"!$\"+++++?!\"*7$$\"+++++'*F-$ \"++++?>F-7$$\"+++++#*F-$\"++++S=F-7$$\"+++++))F-$\"++++gF-7$$FhoF*$! +++++?F--%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBG$\"*++++\"FgwFarFar-F$6%7U 7$F($\"+++++NFgw7$F/$\"++++gLFgw7$F4$\"++++?KFgw7$F9$\"++++!3$Fgw7$F>$ \"++++SHFgw7$FC$F\\qFgw7$FH$\"++++gEFgw7$FM$\"++++?DFgw7$FR$\"++++!Q#F gw7$FW$\"++++SAFgw7$Ffn$\"+++++@Fgw7$F[o$\"++++g>Fgw7$F`o$\"++++?=Fgw7 $Feo$FAFgw7$Fjo$\"++++S:Fgw7$F^p$\"+++++9Fgw7$Fcp$\"++++g7Fgw7$Fgp$F^o Fgw7$F[q$\"+++++)*F-7$F_qF>7$Fcq$\"+++++qF-7$FEF[o7$Fhn$\"+++++UF-7$F` pF[q7$Feq$Fh\\lF-F`r7$Fer$!+++++9F-7$FjrF^t7$F^s$!+++++UF-7$FcsF]v7$Fg s$!+++++qF-7$FjsFiw7$F^t$!+++++)*F-7$Fct$F_vFgw7$Fgt$!++++g7Fgw7$F\\u$ F^^lFgw7$F_u$!++++S:Fgw7$Fdu$F\\xFgw7$Fhu$!++++?=Fgw7$F]v$!++++g>Fgw7$ Fav$!+++++@Fgw7$Fdv$!++++SAFgw7$Fhv$!++++!Q#Fgw7$F]w$!++++?DFgw7$Faw$! ++++gEFgw7$Ffw$F_tFgw7$Fiw$!++++SHFgw7$F^x$!++++!3$Fgw7$Fbx$!++++?KFgw 7$Fgx$!++++gLFgw7$F[y$!+++++NFgwF^yFby-F$6%7U7$F($\"+g#)Q.jF-7$F/$\"+I FD^gF-7$F4$\"+*><\"*z&F-7$F9$\"+p;)pa&F-7$F>$\"+Qh%[H&F-7$FC$\"+31rU]F -7$FH$\"+x]d!z%F-7$FM$\"+Z&R%QXF-7$FR$\"+;SI'G%F-7$FW$\"+'[oT.%F-7$Ffn $\"+cH.#y$F-7$F[o$\"+Du*)HNF-7$F`o$\"+'*=wxKF-7$Feo$\"+kjiDIF-7$Fjo$\" +N3\\tFF-7$F^p$\"+/`N@DF-7$Fcp$\"+u(>#pAF-7$Fgp$\"+WU3#pAF-7$F\\u$F[glF-7$F_u$!+N3\\tFF-7$Fdu $!+kjiDIF-7$Fhu$!+'*=wxKF-7$F]v$!+Du*)HNF-7$Fav$!+cH.#y$F-7$Fdv$!+'[oT .%F-7$Fhv$!+;SI'G%F-7$F]w$!+Z&R%QXF-7$Faw$!+x]d!z%F-7$Ffw$F^glF-7$Fiw$ !+Qh%[H&F-7$F^x$!+p;)pa&F-7$Fbx$!+*><\"*z&F-7$Fgx$!+IFD^gF-7$F[y$!+g#) Q.jF-F^yFby-F$6%7U7$F($\"+-%*zL8Fgw7$F/$\"+EuW!G\"Fgw7$F4$\"+]a4F7Fgw7 $F9$\"+uMut6Fgw7$F>$\"+)\\\"R?6Fgw7$FC$\"+A&Rq1\"Fgw7$FH$\"+Yvo85Fgw7$ FM$\"+&pbLg*F-7$FR$\"+Mf$)p!*F-7$FW$\"+thJO&)F-7$Ffn$\"+8kz-!)F-7$F[o$ \"+_mFpuF-7$F`o$\"+\"*ovNpF-7$Feo$\"+IrB-kF-7$Fjo$\"+ptroeF-7$F^p$\"+3 w>N`F-7$Fcp$\"+[yn,[F-7$Fgp$\"+(3e\"oUF-7$F[q$\"+E$QYt$F-7$F_q$\"+l&=6 ?$F-7$Fcq$\"+/))fnEF-7$FE$\"+V!zS8#F-7$Fhn$\"+#Gf0g\"F-7$F`p$Ff\\mF-7$ Feq$Fd^mFhoF`r7$Fer$!+3w>N`Fho7$Fjr$!+A&Rq1\"F-7$F^s$!+#Gf0g\"F-7$Fcs$ !+V!zS8#F-7$Fgs$!+/))fnEF-7$Fjs$!+l&=6?$F-7$F^t$!+E$QYt$F-7$Fct$!+(3e \"oUF-7$Fgt$!+[yn,[F-7$F\\u$F``mF-7$F_u$!+ptroeF-7$Fdu$!+IrB-kF-7$Fhu$ !+\"*ovNpF-7$F]v$!+_mFpuF-7$Fav$!+8kz-!)F-7$Fdv$!+thJO&)F-7$Fhv$!+Mf$) p!*F-7$F]w$!+&pbLg*F-7$Faw$!+Yvo85Fgw7$Ffw$Fc`mFgw7$Fiw$!+)\\\"R?6Fgw7 $F^x$!+uMut6Fgw7$Fbx$!+]a4F7Fgw7$Fgx$!+EuW!G\"Fgw7$F[y$!+-%*zL8FgwF^yF by-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F[yF(Fbem " 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "SQUARE A CIRCLE " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "To sq uare a circle means to find a square with area equal to a given circle . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 216 "We define a circle at the origin of radius 2. Then we compute the area a nd square root of the area. We construct the square at the origin too. The square has the same area as the circle, and we verify it numerica lly!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " point( orig, 0,0); circle(c1, [orig, 2]); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%origG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#c1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " s := evalf( sqrt(area(c1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG$\"+-x!\\a$! \"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 " square(s1,[point(a , s/2, s/2), point(b, -s/2, s/2), \n point(c, -s/2, -s/2), po int(d, s/2, -s/2)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#s1G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " draw(\{c1,s1(color = coral, filled = true)\} );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7U7$$\"\"#\"\"!$F*F*7$$\"+.%HU)>!\"*$\"+sYm 1D!#57$$\"+Aj;P>F/$\"+Yxzt\\F27$$\"+sHbf=F/$\"+a5\\itF27$$\"+gLh_F/7$$\"+%GEwu$F2$\"+-Xdk>F/7$$\"+&Q5eD\"F2$ \"+dM0'*>F/7$$!+$R5eD\"F2Fbo7$$!+#HEwu$F2$\"+,Xdk>F/7$$!+%))R.='F2Fhn7 $$!+Eee:&)F2FY7$$!+*e`;2\"F/FT7$$!+zz%[F\"F/FO7$$!+ds$zX\"F/$\"+5U4p8F /7$$!+!*R.=;F/$\"+.0dv6F/7$$!+hLh_F/$\"+Ixzt\\F27$$!+.%HU)>F/$\"+gYm1DF27$$!\"#F*$!+I_8/#)! #>7$Fbr$!+wYm1DF27$$!+Aj;P>F/$!+Yxzt\\F27$Fhq$!+a5\\itF27$$!+gLh_F/7$$!+wiiZPF2$!+-Xdk>F/7$$!+x.\"eD\"F2$!+dM0'*>F/7$$\"+,/ \"eD\"F2Fbv7$$\"++jiZPF2$!+,Xdk>F/7$$\"+#*)R.='F2$!+KI6->F/7$$\"+Mee:& )F2$!+0Tl4=F/7$FR$!+^el)o\"F/7$FM$!+'[E5a\"F/7$$\"+ds$zX\"F/$!+4U4p8F/ 7$$\"+\"*R.=;F/$!+-0dv6F/7$$\"+hLh_F/$!+Axzt\\F27$F-$!+_Ym1DF27$F($\"+Yq#3k\"!#=-%&STYLEG6# %%LINEG-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%)POLYGONSG6%7&7$$\"+^QXs< F/F\\[l7$$!+^QXs " 0 "" {MPLTEXT 1 0 28 " area(s1); evalf(area(c1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iqjc7!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iq jc7!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "You can change the radius of the circle and re-execute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 13 "DOUBLE A CUBE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 126 "To double a cube means to find a cube wi th exactly twice the volumne. This is equivalent to constructing the c ube root of two." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 271 "We invoke a new package of commands. Ignore the long lis t of warnings for new definitions. We construct a cube, compute the si de we will need to double it, and then draw both together. Althought i t may not look like it, the larger cube has twice the volume ofthe sma ller!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " with(geom3d):" }}{PARA 7 "" 1 "" {TEXT -1 495 "Warni ng, these names have been redefined: AreCollinear, AreConcurrent, AreC onjugate, AreParallel, ArePerpendicular, DefinedAs, Equation, FindAngl e, GlideReflection, IsEquilateral, IsRightTriangle, OnSegment, Radical Center, altitude, area, center, centroid, circle, coordinates, detail, distance, draw, dsegment, form, homology, homothety, intersection, in version, line, midpoint, point, polar, projection, radius, randpoint, \+ reflection, rotation, segment, sides, translation, triangle, vertices \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " point(orig,0,0,0);\n side_len := 8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%origG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)side_lenG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " cube( cube_1, orig ,side_len/2*sqrt(3) ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " draw( cube_1 , axes = frame d );" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%)POLYG ONSG6(7&7%$\"\"%\"\"!F(F(7%F($!\"%F*F(7%F(F,F,7%F(F(F,7&7%F,F(F(7%F,F( F,7%F,F,F,7%F,F,F(7&F'F/F2F17&F+F4F3F.7&F'F1F4F+7&F.F3F2F/-%(SCALINGG6 #%,CONSTRAINEDG-%*AXESSTYLEG6#%&FRAMEG-%%VIEWG6%;$!+KK?Gp!\"*$\"+KK?Gp FGFDFD" 1 2 0 1 10 0 2 1 1 3 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "evalf( volume( cube _1 ));\nside_2 := evalf( ( volume( cube_1 ))^(1/3) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$7&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' side_2G$\"+++++!)!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " c ube( cube_2, orig, side_2/2*sqrt(3) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " draw( [cube_1, cube_2(style=wireframe, color = blue )],\n axes = framed );" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6(7& 7%$\"\"%\"\"!F(F(7%F($!\"%F*F(7%F(F,F,7%F(F(F,7&7%F,F(F(7%F,F(F,7%F,F, F,7%F,F,F(7&F'F/F2F17&F+F4F3F.7&F'F1F4F+7&F.F3F2F/-F$6*7&7%$\"+******* *R!\"*F=F=7%F=$!+********RF?F=7%F=FAFA7%F=F=FA7&7%FAF=F=7%FAF=FA7%FAFA FA7%FAFAF=7&F