{
  "index": "1991-A-1",
  "type": "GEO",
  "tag": [
    "GEO",
    "ALG"
  ],
  "difficulty": "",
  "question": "A $2 \\times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2,\n3)$. It rotates $90^\\circ$ clockwise about the point $(2, 0)$. It then\nrotates $90^\\circ$ clockwise about the point $(5, 0)$, then $90^\\circ$\nclockwise about the point $(7, 0)$, and finally, $90^\\circ$ clockwise\nabout the point $(10, 0)$. (The side originally on the $x$-axis is now\nback on the $x$-axis.) Find the area of the region above the $x$-axis and\nbelow the curve traced out by the point whose initial position is (1,1).",
  "solution": "Solution.\nfigure 21.\n\nThe point \\( (1,1) \\) rotates around \\( (2,0) \\) to \\( (3,1) \\), then around \\( (5,0) \\) to \\( (6,2) \\), then around \\( (7,0) \\) to \\( (9,1) \\), then around \\( (10,0) \\) to \\( (11,1) \\). (See Figure 21.) The area of concern consists of four \\( 1 \\times 1 \\) right triangles of area \\( 1 / 2 \\), four \\( 1 \\times 2 \\) triangles of area 1 , two quarter circles of area \\( (\\pi / 4)(\\sqrt{2})^{2}=\\pi / 2 \\), and two quarter circles of area \\( (\\pi / 4)(\\sqrt{5})^{2}=5 \\pi / 4 \\), for a total area of \\( 7 \\pi / 2+6 \\).",
  "vars": [
    "x"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "horizontal"
      },
      "question": "A $2 \\times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2,\n3)$. It rotates $90^\\circ$ clockwise about the point $(2, 0)$. It then\nrotates $90^\\circ$ clockwise about the point $(5, 0)$, then $90^\\circ$\nclockwise about the point $(7, 0)$, and finally, $90^\\circ$ clockwise\nabout the point $(10, 0)$. (The side originally on the $horizontal$-axis is now\nback on the $horizontal$-axis.) Find the area of the region above the $horizontal$-axis and\nbelow the curve traced out by the point whose initial position is (1,1).",
      "solution": "Solution.\nfigure 21.\n\nThe point \\( (1,1) \\) rotates around \\( (2,0) \\) to \\( (3,1) \\), then around \\( (5,0) \\) to \\( (6,2) \\), then around \\( (7,0) \\) to \\( (9,1) \\), then around \\( (10,0) \\) to \\( (11,1) \\). (See Figure 21.) The area of concern consists of four \\( 1 \\times 1 \\) right triangles of area \\( 1 / 2 \\), four \\( 1 \\times 2 \\) triangles of area 1 , two quarter circles of area \\( (\\pi / 4)(\\sqrt{2})^{2}=\\pi / 2 \\), and two quarter circles of area \\( (\\pi / 4)(\\sqrt{5})^{2}=5 \\pi / 4 \\), for a total area of \\( 7 \\pi / 2+6 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "dandelion"
      },
      "question": "A $2 \\times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2,\n3)$. It rotates $90^\\circ$ clockwise about the point $(2, 0)$. It then\nrotates $90^\\circ$ clockwise about the point $(5, 0)$, then $90^\\circ$\nclockwise about the point $(7, 0)$, and finally, $90^\\circ$ clockwise\nabout the point $(10, 0)$. (The side originally on the $dandelion$-axis is now\nback on the $dandelion$-axis.) Find the area of the region above the $dandelion$-axis and\nbelow the curve traced out by the point whose initial position is (1,1).",
      "solution": "Solution.\nfigure 21.\n\nThe point \\( (1,1) \\) rotates around \\( (2,0) \\) to \\( (3,1) \\), then around \\( (5,0) \\) to \\( (6,2) \\), then around \\( (7,0) \\) to \\( (9,1) \\), then around \\( (10,0) \\) to \\( (11,1) \\). (See Figure 21.) The area of concern consists of four \\( 1 \\times 1 \\) right triangles of area \\( 1 / 2 \\), four \\( 1 \\times 2 \\) triangles of area 1 , two quarter circles of area \\( (\\pi / 4)(\\sqrt{2})^{2}=\\pi / 2 \\), and two quarter circles of area \\( (\\pi / 4)(\\sqrt{5})^{2}=5 \\pi / 4 \\), for a total area of \\( 7 \\pi / 2+6 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "verticalaxis"
      },
      "question": "A $2 \\times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2,\n3)$. It rotates $90^\\circ$ clockwise about the point $(2, 0)$. It then\nrotates $90^\\circ$ clockwise about the point $(5, 0)$, then $90^\\circ$\nclockwise about the point $(7, 0)$, and finally, $90^\\circ$ clockwise\nabout the point $(10, 0)$. (The side originally on the $verticalaxis$-axis is now\nback on the $verticalaxis$-axis.) Find the area of the region above the $verticalaxis$-axis and\nbelow the curve traced out by the point whose initial position is (1,1).",
      "solution": "Solution.\nfigure 21.\n\nThe point \\( (1,1) \\) rotates around \\( (2,0) \\) to \\( (3,1) \\), then around \\( (5,0) \\) to \\( (6,2) \\), then around \\( (7,0) \\) to \\( (9,1) \\), then around \\( (10,0) \\) to \\( (11,1) \\). (See Figure 21.) The area of concern consists of four \\( 1 \\times 1 \\) right triangles of area \\( 1 / 2 \\), four \\( 1 \\times 2 \\) triangles of area 1 , two quarter circles of area \\( (\\pi / 4)(\\sqrt{2})^{2}=\\pi / 2 \\), and two quarter circles of area \\( (\\pi / 4)(\\sqrt{5})^{2}=5 \\pi / 4 \\), for a total area of \\( 7 \\pi / 2+6 \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp"
      },
      "question": "A $2 \\times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2,\n3)$. It rotates $90^\\circ$ clockwise about the point $(2, 0)$. It then\nrotates $90^\\circ$ clockwise about the point $(5, 0)$, then $90^\\circ$\nclockwise about the point $(7, 0)$, and finally, $90^\\circ$ clockwise\nabout the point $(10, 0)$. (The side originally on the $qzxwvtnp$-axis is now\nback on the $qzxwvtnp$-axis.) Find the area of the region above the $qzxwvtnp$-axis and\nbelow the curve traced out by the point whose initial position is (1,1).",
      "solution": "Solution.\nfigure 21.\n\nThe point \\( (1,1) \\) rotates around \\( (2,0) \\) to \\( (3,1) \\), then around \\( (5,0) \\) to \\( (6,2) \\), then around \\( (7,0) \\) to \\( (9,1) \\), then around \\( (10,0) \\) to \\( (11,1) \\). (See Figure 21.) The area of concern consists of four \\( 1 \\times 1 \\) right triangles of area \\( 1 / 2 \\), four \\( 1 \\times 2 \\) triangles of area 1 , two quarter circles of area \\( (\\pi / 4)(\\sqrt{2})^{2}=\\pi / 2 \\), and two quarter circles of area \\( (\\pi / 4)(\\sqrt{5})^{2}=5 \\pi / 4 \\), for a total area of \\( 7 \\pi / 2+6 \\)."
    },
    "kernel_variant": {
      "question": "A $3\\times4$ rectangle has vertices $(0,0),(3,0),(0,4)$ and $(3,4)$.  Beginning from this position it is subjected to six successive $90^{\\circ}$ clockwise rotations about the points\n$$\\bigl(3,0\\bigr),\\;\\bigl(7,0\\bigr),\\;\\bigl(10,0\\bigr),\\;\\bigl(14,0\\bigr),\\;\\bigl(17,0\\bigr),\\;\\text{and}\\;\\bigl(21,0\\bigr)$$\non the $x$–axis.  Find the area of the region that lies above the $x$–axis and below the curve traced out by the point whose initial position is $(2,2)$.",
      "solution": "Let P₀=(2,2), and successive pivots O₁,…,O₆=(3,0),(7,0),(10,0),(14,0),(17,0),(21,0).  A 90° clockwise rotation sends (x,y)→(y,–x).  Tracking gives\n P₁=(5,1), P₂=(8,2), P₃=(12,2), P₄=(16,2), P₅=(19,1), P₆=(22,2).\nThe six arcs are quarter–circles of radii\n r₁,r₂,r₅,r₆=√5,\n r₃,r₄=√8=2√2,\n so ∑rᵢ²=4·5+2·8=36.\n\nBreak the area under each arc into (a) area under its chord (a trapezoid) plus (b) the circular segment between chord and arc.  For arc i:\n • Trapezoid from xᵢ₋₁ to xᵢ has heights yᵢ₋₁,yᵢ and width Δxᵢ:  Tᵢ=(yᵢ₋₁+yᵢ)/2·Δxᵢ.\n • Segment area = sector minus central triangle = (π/4)rᵢ² – ½rᵢ².\n\nCompute trapezoids:\n Δx’s =3,3,4,4,3,3;  (yᵢ₋₁,yᵢ)=(2,1),(1,2),(2,2),(2,2),(2,1),(1,2)\n ⇒  T’s =4.5,4.5,8,8,4.5,4.5 ⇒ ∑T=34.\n\nSum of segments = ∑[(π/4)rᵢ² – ½rᵢ²] = (π/4–½)·36 = 9π –18.\n\nTotal area = 34 + (9π –18) = 16 + 9π.",
      "_meta": {
        "core_steps": [
          "Track the point’s coordinates after each 90° clockwise rotation about the successive centers.",
          "Realize each rotation moves the point along a quarter-circle whose center is that pivot and whose radius is the point-to-pivot distance at the start of that turn.",
          "Note that the x-axis together with the endpoints of consecutive arcs encloses simple right triangles under every arc.",
          "Compute the radii of the arcs and the legs of the triangles from the coordinate differences obtained in step 1.",
          "Add the areas of all right triangles and quarter-circle sectors to get the required total."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Horizontal side-length of the starting rectangle",
            "original": 2
          },
          "slot2": {
            "description": "Vertical side-length of the starting rectangle",
            "original": 3
          },
          "slot3": {
            "description": "Initial point inside the rectangle (given relative to the lower-left corner)",
            "original": "(1,1)"
          },
          "slot4": {
            "description": "Sequence of rotation centers on the x-axis",
            "original": "[(2,0),(5,0),(7,0),(10,0)]"
          },
          "slot5": {
            "description": "Number of 90° clockwise rotations performed",
            "original": 4
          },
          "slot6": {
            "description": "Radii of the quarter-circle arcs traced out (in the given order)",
            "original": "[√2, √5, √2, √5]"
          },
          "slot7": {
            "description": "Leg lengths of the right triangles formed with the x-axis under successive arcs",
            "original": "1×1 and 1×2"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}