{
  "index": "1999-B-1",
  "type": "GEO",
  "tag": [
    "GEO",
    "ANA"
  ],
  "difficulty": "",
  "question": "Right triangle $ABC$ has right angle at $C$ and $\\angle BAC =\\theta$;\nthe point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$\nis chosen on $BC$ so that $\\angle CDE = \\theta$.  The perpendicular\nto $BC$ at $E$ meets $AB$ at $F$.  Evaluate $\\lim_{\\theta\\rightarrow 0}\n|EF|$.",
  "solution": "The answer is 1/3.\nLet $G$ be the point obtained by reflecting $C$ about the line $AB$.\nSince $\\angle ADC = \\frac{\\pi-\\theta}{2}$, we find that\n$\\angle BDE = \\pi - \\theta - \\angle ADC = \\frac{\\pi-\\theta}{2}\n= \\angle ADC = \\pi - \\angle BDC = \\pi - \\angle BDG$, so that $E,D,G$\nare collinear.  Hence\n\\[\n|EF| = \\frac{|BE|}{|BC|} = \\frac{|BE|}{|BG|} = \\frac{\\sin\n(\\theta/2)}{\\sin (3\\theta/2)},\n\\]\nwhere we have used the law of sines in $\\triangle BDG$.  But by\nl'H\\^opital's Rule,\n\\[\n\\lim_{\\theta \\rightarrow 0}\n\\frac{\\sin(\\theta/2)}{\\sin(3\\theta/2)} =\n\\lim_{\\theta \\rightarrow 0}\n\\frac{\\cos(\\theta/2)}{3\\cos(3\\theta/2)} = 1/3.\n\\]",
  "vars": [
    "A",
    "B",
    "C",
    "D",
    "E",
    "F",
    "G"
  ],
  "params": [
    "\\\\theta"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "A": "vertexa",
        "B": "vertexb",
        "C": "vertexc",
        "D": "vertexd",
        "E": "vertexe",
        "F": "vertexf",
        "G": "vertexg",
        "\\theta": "angletheta"
      },
      "question": "Right triangle $vertexavertexbvertexc$ has right angle at $vertexc$ and $\\angle vertexbvertexavertexc = angletheta$; the point $vertexd$ is chosen on $vertexavertexb$ so that $|vertexavertexc|=|vertexavertexd|=1$; the point $vertexe$ is chosen on $vertexbvertexc$ so that $\\angle vertexcvertexdvertexe = angletheta$.  The perpendicular to $vertexbvertexc$ at $vertexe$ meets $vertexavertexb$ at $vertexf$.  Evaluate $\\lim_{angletheta\\rightarrow 0} |vertexevertexf|$.",
      "solution": "The answer is 1/3.\nLet $vertexg$ be the point obtained by reflecting $vertexc$ about the line $vertexavertexb$.\nSince $\\angle vertexavertexdvertexc = \\frac{\\pi- angletheta}{2}$, we find that\n$\\angle vertexbvertexdvertexe = \\pi - angletheta - \\angle vertexavertexdvertexc = \\frac{\\pi- angletheta}{2}\n= \\angle vertexavertexdvertexc = \\pi - \\angle vertexbvertexdvertexc = \\pi - \\angle vertexbvertexdvertexg$, so that $vertexe,vertexd,vertexg$\nare collinear.  Hence\n\\[\n|vertexevertexf| = \\frac{|vertexbvertexe|}{|vertexbvertexc|} = \\frac{|vertexbvertexe|}{|vertexbvertexg|} = \\frac{\\sin\n(angletheta/2)}{\\sin (3 angletheta/2)},\n\\]\nwhere we have used the law of sines in $\\triangle vertexbvertexdvertexg$.  But by\nl'H\\^opital's Rule,\n\\[\n\\lim_{angletheta \\rightarrow 0}\n\\frac{\\sin(angletheta/2)}{\\sin(3 angletheta/2)} =\n\\lim_{angletheta \\rightarrow 0}\n\\frac{\\cos(angletheta/2)}{3\\cos(3 angletheta/2)} = 1/3.\n\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "A": "lemonade",
        "B": "waterfall",
        "C": "compass",
        "D": "triangle",
        "E": "bicycle",
        "F": "galaxy",
        "G": "notebook",
        "\\theta": "sunflower"
      },
      "question": "Right triangle $lemonade waterfall compass$ has right angle at $compass$ and $\\angle waterfall lemonade compass =sunflower$; the point $triangle$ is chosen on $lemonade waterfall$ so that $|lemonade compass|=|lemonade triangle|=1$; the point $bicycle$ is chosen on $waterfall compass$ so that $\\angle compass triangle bicycle = sunflower$.  The perpendicular to $waterfall compass$ at $bicycle$ meets $lemonade waterfall$ at $galaxy$.  Evaluate $\\lim_{sunflower\\rightarrow 0} |bicycle galaxy|$.",
      "solution": "The answer is 1/3.\nLet $notebook$ be the point obtained by reflecting $compass$ about the line $lemonade waterfall$.\nSince $\\angle lemonade triangle compass = \\frac{\\pi-sunflower}{2}$, we find that\n$\\angle waterfall triangle bicycle = \\pi - sunflower - \\angle lemonade triangle compass = \\frac{\\pi-sunflower}{2} = \\angle lemonade triangle compass = \\pi - \\angle waterfall triangle compass = \\pi - \\angle waterfall triangle notebook$, so that $bicycle,triangle,notebook$ are collinear.  Hence\n\\[\n|bicycle galaxy| = \\frac{|waterfall bicycle|}{|waterfall compass|} = \\frac{|waterfall bicycle|}{|waterfall notebook|} = \\frac{\\sin (sunflower/2)}{\\sin (3sunflower/2)},\n\\]\nwhere we have used the law of sines in $\\triangle$ waterfall triangle notebook.  But by l'H\\^opital's Rule,\n\\[\n\\lim_{sunflower \\rightarrow 0}\n\\frac{\\sin(sunflower/2)}{\\sin(3sunflower/2)} =\n\\lim_{sunflower \\rightarrow 0}\n\\frac{\\cos(sunflower/2)}{3\\cos(3sunflower/2)} = 1/3.\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "A": "voidspace",
        "B": "fullshape",
        "C": "centerless",
        "D": "continuum",
        "E": "surfaceless",
        "F": "boundaryless",
        "G": "infinitude",
        "\\theta": "straightness"
      },
      "question": "Right triangle $voidspacefullshapecenterless$ has right angle at $centerless$ and $\\angle fullshapevoidspacecenterless = straightness$; the point $continuum$ is chosen on $voidspacefullshape$ so that $|voidspacecenterless|=|voidspacecontinuum|=1$; the point $surfaceless$ is chosen on $centerlessfullshape$ so that $\\angle centerlesscontinuumsurfaceless = straightness$.  The perpendicular to $centerlessfullshape$ at $surfaceless$ meets $voidspacefullshape$ at $boundaryless$.  Evaluate $\\lim_{straightness\\rightarrow 0}\n|surfacelessboundaryless|$.",
      "solution": "The answer is 1/3.\nLet $infinitude$ be the point obtained by reflecting $centerless$ about the line $voidspacefullshape$.\nSince $\\angle voidspacecontinuumcenterless = \\frac{\\pi-straightness}{2}$, we find that\n$\\angle fullshapecontinuumsurfaceless = \\pi - straightness - \\angle voidspacecontinuumcenterless = \\frac{\\pi-straightness}{2}\n= \\angle voidspacecontinuumcenterless = \\pi - \\angle fullshapecontinuumcenterless = \\pi - \\angle fullshapecontinuuminfinitude$, so that $surfaceless,continuum,infinitude$\nare collinear.  Hence\n\\[\n|surfacelessboundaryless| = \\frac{|fullshapesurfaceless|}{|centerlessfullshape|} = \\frac{|fullshapesurfaceless|}{|fullshapeinfinitude|} = \\frac{\\sin\n(straightness/2)}{\\sin (3straightness/2)},\n\\]\nwhere we have used the law of sines in $\\triangle fullshapecontinuuminfinitude$.  But by\nl'H\\^opital's Rule,\n\\[\n\\lim_{straightness \\rightarrow 0}\n\\frac{\\sin(straightness/2)}{\\sin(3straightness/2)} =\n\\lim_{straightness \\rightarrow 0}\n\\frac{\\cos(straightness/2)}{3\\cos(3straightness/2)} = 1/3.\n\\]"
    },
    "garbled_string": {
      "map": {
        "A": "qfhxmbce",
        "B": "aztdwkrn",
        "C": "nslvpkqo",
        "D": "wxyorjdf",
        "E": "ghqbtils",
        "F": "kcdyjpru",
        "G": "ovmzreal",
        "\\theta": "pxgcthru"
      },
      "question": "Right triangle $qfhxmbceaztdwkrnnslvpkqo$ has right angle at $nslvpkqo$ and $\\angle aztdwkrnqfhxmbcenslvpkqo = pxgcthru$; the point $wxyorjdf$ is chosen on $qfhxmbceaztdwkrn$ so that $|qfhxmbcenslvpkqo|=|qfhxmbcewxyorjdf|=1$; the point $ghqbtils$ is chosen on $aztdwkrnnslvpkqo$ so that $\\angle nslvpkqowxyorjdfghqbtils = pxgcthru$.  The perpendicular to $aztdwkrnnslvpkqo$ at $ghqbtils$ meets $qfhxmbceaztdwkrn$ at $kcdyjpru$.  Evaluate $\\lim_{pxgcthru\\rightarrow 0} |ghqbtilskcdyjpru|$.",
      "solution": "The answer is 1/3.\nLet $ovmzreal$ be the point obtained by reflecting $nslvpkqo$ about the line $qfhxmbceaztdwkrn$.\nSince $\\angle qfhxmbcewxyorjdfnslvpkqo = \\frac{\\pi-pxgcthru}{2}$, we find that\n$\\angle aztdwkrnwxyorjdfghqbtils = \\pi - pxgcthru - \\angle qfhxmbcewxyorjdfnslvpkqo = \\frac{\\pi-pxgcthru}{2}\n= \\angle qfhxmbcewxyorjdfnslvpkqo = \\pi - \\angle aztdwkrnnslvpkqowxyorjdf = \\pi - \\angle aztdwkrnwxyorjdfovmzreal$, so that $ghqbtils,wxyorjdf,ovmzreal$\nare collinear.  Hence\n\\[\n|ghqbtilskcdyjpru| = \\frac{|aztdwkrnghqbtils|}{|aztdwkrnnslvpkqo|} = \\frac{|aztdwkrnghqbtils|}{|aztdwkrnovmzreal|} = \\frac{\\sin\n(pxgcthru/2)}{\\sin (3pxgcthru/2)},\n\\]\nwhere we have used the law of sines in $\\triangle aztdwkrnwxyorjdfovmzreal$.  But by\nl'H\\^opital's Rule,\n\\[\n\\lim_{pxgcthru \\rightarrow 0}\n\\frac{\\sin(pxgcthru/2)}{\\sin(3pxgcthru/2)} =\n\\lim_{pxgcthru \\rightarrow 0}\n\\frac{\\cos(pxgcthru/2)}{3\\cos(3pxgcthru/2)} = 1/3.\n\\]"
    },
    "kernel_variant": {
      "question": "In right-triangle $ABC$ the right angle is at $C$ and $\\angle BAC = \\theta\\;(0<\\theta<\\tfrac{\\pi}{2})$.  A point $D$ on $AB$ is chosen so that $|AC| = |AD| = 3$.  Next choose the point $E$ on $BC$ satisfying $\\angle CDE = \\theta$.  Let $F$ be the intersection of $AB$ with the line through $E$ perpendicular to $BC$ (so $EF \\perp BC$).\n\nEvaluate\n\\[\\displaystyle\\lim_{\\theta\\to 0}|EF|.\\]",
      "solution": "Step 1.  Choosing convenient coordinates\nBecause only ratios of lengths matter in the limit, we first work with the similar triangle that has $|AC| = |AD| = 1$ and later scale the result by 3.\n\nPlace\nA = $(0,0)$ so that $AB$ lies on the $x$-axis.\nSince $|AC|=1$ and $\\angle CAB = \\theta$, we may take\nC = $(\\cos\\theta,\\,\\sin\\theta)$.\nThe right angle is at $C$, so the vector $\\overrightarrow{CB}$ must be perpendicular to $\\overrightarrow{CA}$.  Writing $B=(x_B,0)$ and imposing $(A-C)\\cdot(B-C)=0$ gives\n\\[( -\\cos\\theta,\\,-\\sin\\theta)\\cdot (x_B-\\cos\\theta,\\,-\\sin\\theta)=0 \\Longrightarrow x_B=\\sec\\theta.\\]\nThus $B=(\\sec\\theta,0)$ (note that $B$ is \nnot the foot of the altitude from $C$ to $AB$; it is the second vertex that makes $\\triangle ABC$ right at $C$).\n\nBecause $|AD|=1=|AC|$, the point $D$ is $(1,0)$.\n\nStep 2.  Locating $E$\nReflect $C$ across $AB$ to obtain $G=(\\cos\\theta,-\\sin\\theta)$.  Since $AC=AD$, $\\triangle ACD$ is isosceles and\n\\[\\angle ADC = \\angle ACD = \\frac{\\pi-\\theta}{2}.\\]\nA routine angle chase then shows that $\\angle BDE = \\angle BDG$, so $D,E,G$ are collinear.  Hence $E$ is the intersection of lines $DG$ and $BC$.\n\nParametric equations:\nDG: $D+u(G-D)=(1+u(\\cos\\theta-1),\\,-u\\sin\\theta)$,\nBC: $B+v(C-B)=(\\sec\\theta+v(\\cos\\theta-\\sec\\theta),\\,v\\sin\\theta)$.\nEqual $y$-coordinates give $-u\\sin\\theta=v\\sin\\theta\\Rightarrow v=-u$.  Substituting into the $x$-coordinate yields\n\\[1+u(\\cos\\theta-1)=\\sec\\theta-u(\\cos\\theta-\\sec\\theta)\\Longrightarrow u=-\\frac{1}{2\\cos\\theta+1}.\\]\nTherefore\n\\[E=\\Bigl(\\tfrac{2+\\cos\\theta}{2\\cos\\theta+1},\\;\\tfrac{\\sin\\theta}{2\\cos\\theta+1}\\Bigr).\\]\n\nStep 3.  The length $EF$\nLine $BC$ has slope $-(\\cot\\theta)$, hence the line through $E$ perpendicular to $BC$ has slope $\\tan\\theta$.  Its equation is\n\\[y-\\frac{\\sin\\theta}{2\\cos\\theta+1}=\\tan\\theta\\Bigl(x-\\frac{2+\\cos\\theta}{2\\cos\\theta+1}\\Bigr).\\]\nSetting $y=0$ (because $F$ lies on $AB$) is unnecessary: the vertical drop of $E$ to the $x$-axis together with the inclination of the perpendicular already gives\n\\[|EF|=\\frac{\\text{vertical distance of }E\\text{ from }AB}{\\sin(\\angle(\\text{perpendicular},AB))}\n          =\\frac{\\dfrac{\\sin\\theta}{2\\cos\\theta+1}}{\\sin\\theta}\n          =\\frac{1}{2\\cos\\theta+1}.\\]\n\nStep 4.  Taking the limit for the unit triangle\n\\[\\lim_{\\theta\\to 0}|EF| = \\frac{1}{2\\cdot 1+1}=\\frac13.\\]\n\nStep 5.  Restoring the original scale\nAll lengths in the configuration scale linearly with $|AC|$.  Because the original problem has $|AC|=3$, multiply by 3:\n\\[\\boxed{\\displaystyle\\lim_{\\theta\\to 0}|EF| = 3\\cdot\\frac13 = 1}.\\]",
      "_meta": {
        "core_steps": [
          "Reflect C across AB to create point G",
          "Angle-chase to prove E, D, G are collinear",
          "EF ∥ AC  ⇒  |EF|/|AC| = |BE|/|BC| by similarity",
          "Law of sines in ΔBDG gives |BE|/|BC| = sin(θ/2)/sin(3θ/2)",
          "Apply L’Hôpital as θ → 0 to obtain the limit"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Numerical value chosen for the common length AC = AD; only the equality matters, not the fact it equals 1.",
            "original": "1"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation",
  "iteratively_fixed": true
}