{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260912-distance-coordinate-0018",
  "pilot_number": "P250-0018",
  "number": 18,
  "family_id": "qf-distance-coordinate",
  "title": {
    "zh-Hant": "第 18 題：座標平面距離計算",
    "en": "Question 18: Coordinate-Plane Distance"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260912-distance-coordinate-0018",
    "batch_id": "batch-20260912-geometry-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euclidean-length"
    ],
    "title_zh": "座標平面距離計算",
    "stem_zh": "在歐幾里得平面上，點 P=(2, 2) 與 Q=(-58, 93)。求距離 d(P,Q)。",
    "answer": {
      "kind": "numeric",
      "value": 109,
      "explanation_zh": "座標差為 (-60, 91)，所以 d=√((-60)²+91²)=109。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "座標差",
        "距離公式"
      ]
    },
    "variant": {
      "family_id": "qf-distance-coordinate",
      "seed": 20260929,
      "parameters": {
        "p": [
          2,
          2
        ],
        "q": [
          -58,
          93
        ],
        "expected_distance": 109
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "schema_pending",
        "independent_recomputation_pending"
      ],
      "warning_zh": "本題僅對列出的有限參數重算；不構成相關定義或定理的普遍證明，且尚未人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-09-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260912-distance-coordinate-0018",
    "batch_id": "batch-20260912-geometry-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
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      "family_id": "qf-distance-coordinate",
      "seed": 20260929,
      "semantic_template": "qf-distance-coordinate/20260929"
    },
    "fields": {
      "title": "Coordinate-Plane Distance",
      "stem": "In the Euclidean plane, P=(2, 2) and Q=(-58, 93). Find the distance d(P,Q).",
      "answer": {
        "kind": "numeric",
        "value": 109,
        "explanation": "The coordinate difference is (-60, 91), so d=√((-60)²+(91)²)=109."
      },
      "skills": [
        "coordinate differences",
        "distance formula"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "在歐幾里得平面上，點 P=(2, 2) 與 Q=(-58, 93)。求距離 d(P,Q)。",
        "en": "In the Euclidean plane, P=(2, 2) and Q=(-58, 93). Find the distance d(P,Q)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "P=(2,2), Q=(-58,93)",
      "after": "Δx=Q_x−P_x; Δy=Q_y−P_y",
      "relation": "implies",
      "rule_id": "coordinate_difference",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先固定相同方向 Q−P，分別求兩個座標差。",
        "en": "Use the same direction Q−P for both coordinate differences."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "交換兩點會同時改變差的符號，但不改變平方和。",
          "en": "Swapping the points changes both signs but not the sum of squares."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Δx := sub(-58, 2)",
      "after": "Δx = -60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "橫座標差是 -58 減去 2，負座標仍須保留括號。",
        "en": "Subtract 2 from -58; keep parentheses around negative coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Δx，不以小數近似替代。",
          "en": "Recompute Δx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-58",
          "2"
        ],
        "output": "-60",
        "label": "Δx"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Δy := sub(93, 2)",
      "after": "Δy = 91",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "縱座標差是 93 減去 2。",
        "en": "The vertical difference is 93 minus 2."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Δy，不以小數近似替代。",
          "en": "Recompute Δy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "93",
          "2"
        ],
        "output": "91",
        "label": "Δy"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "(Δx)² := square(-60)",
      "after": "(Δx)² = 3600",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把完整的橫差平方，不能只平方減數。",
        "en": "Square the entire horizontal difference, not just the subtracted coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 (Δx)²，不以小數近似替代。",
          "en": "Recompute (Δx)² with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "-60"
        ],
        "output": "3600",
        "label": "(Δx)²"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "(Δy)² := square(91)",
      "after": "(Δy)² = 8281",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "縱差即使為負，平方後仍非負。",
        "en": "The square is nonnegative even when the vertical difference is negative."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 (Δy)²，不以小數近似替代。",
          "en": "Recompute (Δy)² with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "91"
        ],
        "output": "8281",
        "label": "(Δy)²"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "d² := sum(3600, 8281)",
      "after": "d² = 11881",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依歐幾里得距離公式加總兩個平方。",
        "en": "Add the two squares in the Euclidean distance formula."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 d²，不以小數近似替代。",
          "en": "Recompute d² with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3600",
          "8281"
        ],
        "output": "11881",
        "label": "d²"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "d²=11881, d≥0",
      "after": "d=√(11881)=109",
      "relation": "implies",
      "rule_id": "nonnegative_square_root",
      "conditions": [
        "d≥0"
      ],
      "explanation": {
        "zh-Hant": "因為 109²=11881 且距離不為負，取非負平方根 109。",
        "en": "Since 109²=11881 and distance is nonnegative, choose the nonnegative root 109."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "負平方根不是距離；不將平方距離直接當作距離。",
          "en": "The negative root is not a distance, and squared distance is not distance itself."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "answer² := square(109)",
      "after": "answer² = 11881",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把所得距離平方，回核原來的平方和。",
        "en": "Square the proposed distance to recover the original sum of squares."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 answer²，不以小數近似替代。",
          "en": "Recompute answer² with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "109"
        ],
        "output": "11881",
        "label": "answer²"
      }
    }
  ],
  "math_model": {
    "dx": "-60",
    "dy": "91",
    "square": "11881",
    "root": 109
  },
  "conclusion": {
    "canonical_value": 109,
    "explanation": {
      "zh-Hant": "兩點距離為 109。",
      "en": "The distance is 109."
    }
  },
  "definition_note": {
    "zh-Hant": "距離取非負主平方根，沿用引用來源既有假設 a3；相同點的距離為零。本批兩點均不同，結果皆為正。",
    "en": "Distance uses the nonnegative principal square root, following existing source assumption a3. Coincident points have zero distance; the selected pairs are distinct and have positive distances."
  },
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      "en": "This is a source-bound pedagogical derivation candidate. Exact recomputation is not independent mathematical review or formal proof. Original questions, answers, formulas, ASTs and history are unchanged."
    },
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  },
  "revision": {
    "id": "review-20261024-001",
    "review_date": "2026-10-24",
    "owner_authorized_reader_publication": true,
    "scope": "Independent second-context reading of all25 bilingual derivations and200 steps, with question-first exact recalculation. Not human review, a formal proof, whole36000 acceptance or Canonical promotion.",
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}