{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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 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 1 128 0 0 1 0 1 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 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 285 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 128 0 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 "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 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 51 "High School Modul es > Geometry by Gregory A. Moore\n" }}{PARA 3 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 256 25 "The Theorem of Pythagoras" }}{PARA 0 "" 0 "" {TEXT -1 78 "\nAn exploration of this famous and fundamental theorem, and it s applications.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Exec ute the Code Resource section first. Although there will be no output \+ immediately, these definitions are used 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 22 "r estart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "c ef := 'color=COLOR(RGB,.7,.7,.4), filled = true':\ncbf := 'color=COLOR (RGB,.4,.4,.8), filled = true':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 481 "RtTriPlot := proc(A,B,C)\nlocal sA,sB,sC,d;\n\nsA := [ (C[1]+ B[1])/2, (C[2]+B[2])/2];\nsB := [ (A[1]+C[1])/2, (A[2]+C[2])/2];\nsC : = [ (A[1]+B[1])/2, (A[2]+B[2])/2];\nd := .6:\ndisplay(\n plot( [A,B ,C,A], cbf,\n thickness = 3, axes = none, scaling = constrained ),\n textplot( [sA[1], sA[2]-d, `a`],font=[TIMES,ROM AN,20]),\n textplot( [sB[1]-d, sB[2], `b`],font=[TIMES,ROMAN,20]) ,\n textplot( [sC[1]+d, sC[2]+d, `c`],font=[TIMES,ROMAN,20]) \n \+ );\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "RtTriPl ot2 := proc(a,b)\nlocal A,B,C,c, sA,sB,sC,d;\nB := [a,0]; A:=[0,b]; C: = [0,0];\nsB := [ 0, b/2];\nsA := [ a/2, 0];\nsC := [ a/2, b/2];\nc := sqrt( a^2 + b^2);\nd := .3:\ndisplay(\n plot( [A,B,C,A], cef,\n \+ thickness = 3, axes = none, scaling = constrained ), \n textplot( [sA[1], sA[2]-d, a],font=[TIMES,ROMAN,15]),\n tex tplot( [sB[1]-d, sB[2], b],font=[TIMES,ROMAN,15]),\n textplot( [s C[1]+d, sC[2]+d, c],font=[TIMES,ROMAN,15]) \n );\nend:\n" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 14 "1. The Theorem" }}{PARA 0 "" 0 "" {TEXT -1 54 "\nWe all know what a right triangle loo ks like, right?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "RtTriPl ot( [0,8],[6,0],[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 13 "\n\nOr this \+ :\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "RtTriPlot( [0,5],[12, 0],[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 46 "\n\nThe theorem is this : \+ (trumpets please....)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a ^2 + b^2 = c^2;" }}}{PARA 0 "" 0 "" {TEXT -1 47 "\nAlthough it sometim es takes on this disguise :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c = sqrt( a^2 + b^2);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 12 "2. The Proof" }} {PARA 0 "" 0 "" {TEXT -1 200 "\nThere are over 100 proofs that have be en devised over the past two millennia for this fundamental theorem. W e will create a famous diagram which can be an aid in proving the vali dity of the theorem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a := 5; b := 7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "display( 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 0 "" 0 "" {TEXT -1 105 "\nHow does this prove the theorem? Wel l we compute the area for this diagram in two different ways.\n\n \+ " }{TEXT 265 17 "1. Area of Pieces" }{TEXT -1 84 "\n\nEach of the four triangles (green, red, blue, purple) has dimensions a, b, and c.\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a:= 'a': b:= 'b':\n`Area of each triangle` = (1/2)*a*b; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "`Area of all four triangles` = 4*(1/2)*a*b;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "`Area of Yellow Square in the middle` = c^2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "`Total Area of Diagram` = %% + %;" }}}{PARA 0 "" 0 "" {TEXT -1 7 " \n " }{TEXT 266 13 "2. Total Area" }{TEXT -1 248 "\n\nWe can also compute the area in a diff erent way. The entire diagram is actually a square. The length of the \+ side of the big square is a + b, since each side of the square consist s one small side of the triangle and one larger leg of the triangle." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "`Total Area of Diagram` = \+ (a+b)^2;" }}}{PARA 0 "" 0 "" {TEXT -1 10 " \n " }{TEXT 267 30 " Equating the Two Types of Area" }{TEXT -1 71 "\n\nNow we equate the tw o different computations for the area, and solve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "2*a*b + c^2 = (a+b)^2;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "lhs(%)-rhs(%) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c^2 = solve(%, c^2);" }}}{PARA 0 "" 0 "" {TEXT -1 83 "\n Voila!...oops....I mean, \"QED\" (quo est demonstratum...thus it i s demonstrated).\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 263 39 "3. Finding the Sides of Right Triangles" }}{PARA 0 "" 0 "" {TEXT -1 317 "\nGiven any two sides of a right triangle its possible t o find the third side using this formula. The one key issue about usin g the formula is to substitute into it properly. Just remember that c, the hypotenuse, is the longest side of the triangle, and its square i s equal to the sum of the squares of the other two.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Pyth := c^2 = a^2+b^2;" }}}{PARA 0 "" 0 " " {TEXT -1 2 "\n\n" }{TEXT 268 20 " Example 3.1:" }{TEXT 271 64 " A right triangle has legs of side 5 and 12, find the hypotenuse" }{TEXT -1 76 "\n\nIn this case, we are given the two legs - the smalle r two sides, a and b.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s ubs( \{a = 5, b = 12\}, Pyth);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "c = 'sqrt(169)'; \nexpand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "RtTriPlot2(5,12);" }}}{PARA 0 "" 0 "" {TEXT -1 107 " \nThat worked nicely to an integer, but usually, there is a square roo t involved in one way or another.\n\n\n \n" }{TEXT 274 15 " Exa mple" }{TEXT 275 1 " " }{TEXT 276 4 "3.2:" }{TEXT 277 63 " A right tri angle has legs of side 2 and 3, find the hypotenuse" }{TEXT -1 76 "\n \nIn this case, we are given the two legs - the smaller two sides, a a nd b.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs( \{a = 2, b \+ = 3\}, Pyth);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c = sqrt(1 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "RtTriPlot2(2,3);" }} }{PARA 0 "" 0 "" {TEXT -1 69 "\nHere is the other variation - given th e hypotenuse and one leg. \n\n\n\n" }{TEXT 269 15 " Example" } {TEXT 270 1 " " }{TEXT 272 4 "3.3:" }{TEXT 273 87 " A right triangle h as leg 10 inches long and hypotenuse 15 inches, find the other side." }{TEXT -1 76 "\n\nIn this case, we are given the two legs - the smalle r two sides, a and b.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "s ubs( \{a = 10, c = 15\}, Pyth);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b^2 = solve(%, b^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "b = 'sqrt(125)'; \nexpand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "RtTriPlot2( 10, 5*sqrt(5) );" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 264 29 "4. Well Known Right Triangles" }}{PARA 0 "" 0 "" {TEXT -1 91 " \nThere are two very common families of right triangles, which are wel l worth remembering.\n\n" }{TEXT 278 25 " 45-45-90 Triangle" } {TEXT -1 85 "\n\nWhen the two legs are equal, the two base angles are \+ equal also - 45 degrees each.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RtTriPlot2( 1, 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs( \{a = x, b = x\}, Pyth);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "c = sqrt( 2*x^2);\nc = x*sqrt(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "RtTriPlot2(7,7);" }}}{PARA 0 "" 0 "" {TEXT -1 26 "\nThe hypotenuse is always " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqr tG6#\"\"#" }{TEXT -1 110 " times the side of the either of the two leg s on a 45-45-90 triangle.\n\n\n\n The other common triangle is this.\n \n" }{TEXT 279 25 " 30-60-90 Triangle" }{TEXT -1 232 "\n\nThis \+ kind of triangle can be considered to be half of an equilateral triang le, and so the shorter base is half the hypotenuse. Knowing this much \+ allows us to express the third side (the longer leg) in terms of the o ther two also.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "RtTriPlo t2(1,sqrt(3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "RtTriPlot 2( sqrt(3)/2, 1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs ( \{a = x, c = 2*x\}, Pyth);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b^2 = solve(%, b^2);" }}}{PARA 0 "" 0 "" {TEXT -1 55 "\nThis giv es us this relationship among the sides : " }{TEXT 280 4 "1 : " } {XPPEDIT 281 0 "sqrt(3)" "6#-%%sqrtG6#\"\"$" }{TEXT 282 4 " : 2" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a = x;\nb = x*sqrt(3);\nc = \+ 2*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Tri369 := \{a = x, b = x*sqrt(3), c = 2*x \};" }}}{PARA 0 "" 0 "" {TEXT -1 91 "\nOften w e see this kind of triangle expressed in this form, where the hypotenu se is 1. " }{TEXT 284 1 " " }{XPPEDIT 285 0 "1/2" "6#*&\"\"\"F$\" \"#!\"\"" }{TEXT 286 3 " : " }{XPPEDIT 257 0 "sqrt(3)/2;" "6#*&-%%sqrt G6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 287 3 " : " }{TEXT 283 1 "1" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(x = 1/2, Tri369);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "RtTriPlot2(7,7*sqrt(3));" }} }{EXCHG }{EXCHG }{PARA 0 "" 0 "" {TEXT -1 99 "\nPlease take a look at \+ the \"Pythagorean Triples\" worksheet for some interesting related mat erial.\n " }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 2002 Water loo Maple Inc " }}}{MARK "0 1" 37 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }