{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0001","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aa5db442b49ef918d38dfc544d3616b465865cfe2a05a19ee51d87a304f4903","family_id":"qf-transform-translation-point","seed":202610077001,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(0, 0) and translation vector t=[0, 0]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Add the two components of the specified translation, obtaining the exact image (0, 0). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0002","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57893d35679580f2703a37b1b2c471516d5bff386c4616a18be2813fcd211546","family_id":"qf-transform-translation-point","seed":202610077002,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-11\",\"0\"],[\"0\",\"0\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-11, 0) and translation vector t=[0, 0]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-11, 0)","explanation":"Add the two components of the specified translation, obtaining the exact image (-11, 0). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0003","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7021bb850b520e2b8347d824c4d1dc9a7c62ed5cd30122b313c1efbbf6b13ca8","family_id":"qf-transform-translation-point","seed":202610077003,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-10\",\"-1\"],[\"1\",\"2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-10, -1) and translation vector t=[1, 2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-9, 1)","explanation":"Add the two components of the specified translation, obtaining the exact image (-9, 1). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0004","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36c16ee8d0970568a81665f68da9eeca6c27945b244e9a4de563d2a1cdfbf818","family_id":"qf-transform-translation-point","seed":202610077004,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-9\",\"-2\"],[\"-1\",\"-1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-9, -2) and translation vector t=[-1, -1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-10, -3)","explanation":"Add the two components of the specified translation, obtaining the exact image (-10, -3). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0005","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6eb55194b5fa0e6519debee0bdf99212219c6cf5774b95b03903c518116bf49","family_id":"qf-transform-translation-point","seed":202610077005,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-8\",\"-3\"],[\"0\",\"1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-8, -3) and translation vector t=[0, 1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-8, -2)","explanation":"Add the two components of the specified translation, obtaining the exact image (-8, -2). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0006","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d915506c2f5e204824c459604ae67bcd3a6c9dec37139cc2c0ee86cc68d4c347","family_id":"qf-transform-translation-point","seed":202610077006,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-7\",\"-4\"],[\"1\",\"-2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-7, -4) and translation vector t=[1, -2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-6, -6)","explanation":"Add the two components of the specified translation, obtaining the exact image (-6, -6). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0007","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"422ec125de7318ebed11879a0234b84a82325dcbe5e6796faeab38563d1e447c","family_id":"qf-transform-translation-point","seed":202610077007,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-6\",\"-5\"],[\"-1\",\"0\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-6, -5) and translation vector t=[-1, 0]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-7, -5)","explanation":"Add the two components of the specified translation, obtaining the exact image (-7, -5). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0008","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaa730a505fb3ed46ae90f8eb3a47e54e03888c0d2166e21478711a0ed584715","family_id":"qf-transform-translation-point","seed":202610077008,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-5\",\"-6\"],[\"0\",\"2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-5, -6) and translation vector t=[0, 2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-5, -4)","explanation":"Add the two components of the specified translation, obtaining the exact image (-5, -4). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0009","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a384049dd7c879fc2e1722d8a2e9cf1528f38ea1a9e1dcc1bde878154965c26e","family_id":"qf-transform-translation-point","seed":202610077009,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-4\",\"-7\"],[\"1\",\"-1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-4, -7) and translation vector t=[1, -1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-3, -8)","explanation":"Add the two components of the specified translation, obtaining the exact image (-3, -8). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0010","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13ea4ec62d7001080fb5dcf1e6a65619d32d35ee9e9be260f9d9b75fbab1cd0c","family_id":"qf-transform-translation-point","seed":202610077010,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-3\",\"-8\"],[\"-1\",\"1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-3, -8) and translation vector t=[-1, 1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-4, -7)","explanation":"Add the two components of the specified translation, obtaining the exact image (-4, -7). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0011","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3160a3654609ca342c4a884376b4b0eb4e9e899892ab68cd4f26130253f9eef5","family_id":"qf-transform-translation-point","seed":202610077011,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-2\",\"-9\"],[\"0\",\"-2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-2, -9) and translation vector t=[0, -2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-2, -11)","explanation":"Add the two components of the specified translation, obtaining the exact image (-2, -11). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0012","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dbaf112f771a4e60a7aadf8746c1056a50011044386ce3f3975e758876118bb","family_id":"qf-transform-translation-point","seed":202610077012,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-1\",\"-10\"],[\"1\",\"0\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-1, -10) and translation vector t=[1, 0]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(0, -10)","explanation":"Add the two components of the specified translation, obtaining the exact image (0, -10). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0013","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab7976bf57c9a297939e6ec0838245652dd97958451d6f03bc9b224f8da9a13","family_id":"qf-transform-translation-point","seed":202610077013,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"0\",\"-11\"],[\"-1\",\"2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(0, -11) and translation vector t=[-1, 2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(-1, -9)","explanation":"Add the two components of the specified translation, obtaining the exact image (-1, -9). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0014","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d408376c0b7a7548c417360b09672ee8a560dfa0c62b79270f1cf299e8ce72a","family_id":"qf-transform-translation-point","seed":202610077014,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"1\",\"-12\"],[\"0\",\"-1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(1, -12) and translation vector t=[0, -1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(1, -13)","explanation":"Add the two components of the specified translation, obtaining the exact image (1, -13). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0015","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"635ea081e601c12dc8b8bb420db6c5f08afb3cf49b70c4bf24e364f7a9ea9989","family_id":"qf-transform-translation-point","seed":202610077015,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"2\",\"-13\"],[\"1\",\"1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(2, -13) and translation vector t=[1, 1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(3, -12)","explanation":"Add the two components of the specified translation, obtaining the exact image (3, -12). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0016","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12a4d7036079b0ed9e15d6b438fdf103e3966d7ba488e7dc00d2e315efa97389","family_id":"qf-transform-translation-point","seed":202610077016,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"3\",\"-14\"],[\"-1\",\"-2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(3, -14) and translation vector t=[-1, -2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(2, -16)","explanation":"Add the two components of the specified translation, obtaining the exact image (2, -16). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0017","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"785207e8c42eb862bb4becc46758b2578bc3a7b3793fbfe3da6d038eaf1d27d4","family_id":"qf-transform-translation-point","seed":202610077017,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"4\",\"-15\"],[\"0\",\"0\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(4, -15) and translation vector t=[0, 0]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(4, -15)","explanation":"Add the two components of the specified translation, obtaining the exact image (4, -15). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0018","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dba0255a16525739d99237c55118b2d210c26e6b7c8bbc5c61ff8f1fc3f2d2e9","family_id":"qf-transform-translation-point","seed":202610077018,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"5\",\"-16\"],[\"1\",\"2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(5, -16) and translation vector t=[1, 2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(6, -14)","explanation":"Add the two components of the specified translation, obtaining the exact image (6, -14). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0019","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ed180c97025d9bb817ec2e39bf788ba713e8b5407881c67f29aefde61705829","family_id":"qf-transform-translation-point","seed":202610077019,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"6\",\"-17\"],[\"-1\",\"-1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(6, -17) and translation vector t=[-1, -1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(5, -18)","explanation":"Add the two components of the specified translation, obtaining the exact image (5, -18). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0020","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bfec5fc8a24b2ca0d35823a226d1f6de0514620d4e35fa6d50fcbddb69dae2f","family_id":"qf-transform-translation-point","seed":202610077020,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"7\",\"-18\"],[\"0\",\"1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(7, -18) and translation vector t=[0, 1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(7, -17)","explanation":"Add the two components of the specified translation, obtaining the exact image (7, -17). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0021","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d7eae30c7a0e6861a6d4087ce7b46a24d1cb7cc39176ca112921ccb72fb5ea3","family_id":"qf-transform-translation-point","seed":202610077021,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"8\",\"-19\"],[\"1\",\"-2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(8, -19) and translation vector t=[1, -2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(9, -21)","explanation":"Add the two components of the specified translation, obtaining the exact image (9, -21). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0022","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c7b1e8971ca4f7bbd271b005932ec5875f4d874afc8396370299fbbf35ba3fd","family_id":"qf-transform-translation-point","seed":202610077022,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"9\",\"-20\"],[\"-1\",\"0\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(9, -20) and translation vector t=[-1, 0]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(8, -20)","explanation":"Add the two components of the specified translation, obtaining the exact image (8, -20). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0023","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f495221b5d1029cf740855294c26fb170e78bc4910b653480cf9ddfa7ca92ebf","family_id":"qf-transform-translation-point","seed":202610077023,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"10\",\"-21\"],[\"0\",\"2\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(10, -21) and translation vector t=[0, 2]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(10, -19)","explanation":"Add the two components of the specified translation, obtaining the exact image (10, -19). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0024","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afbac05caf83530a8f53f150016e0945c5699fe4556cee42f3ed89863e654855","family_id":"qf-transform-translation-point","seed":202610077024,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"11\",\"-22\"],[\"1\",\"-1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(11, -22) and translation vector t=[1, -1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(12, -23)","explanation":"Add the two components of the specified translation, obtaining the exact image (12, -23). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0025","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab35cd35326699ddf732fcd26830ce8b66503cd860de3793a46ce189d73d9aeb","family_id":"qf-transform-translation-point","seed":202610077025,"semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"12\",\"-23\"],[\"-1\",\"1\"]],\"operation\":\"translation_point\"}"},"fields":{"title":"Exact Image Point under Translation","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(12, -23) and translation vector t=[-1, 1]. Find the coordinates of P+t.","answer":{"kind":"expression","value":"(11, -22)","explanation":"Add the two components of the specified translation, obtaining the exact image (11, -22). Retain the actual image position rather than only a difference between points."},"skills":["Exact Image Point under Translation"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0026","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"766c6b8e9c7a0375705a218bfe5ce816330dc7a537dc6ec7777658e7612dbcd9","family_id":"qf-transform-quarter-turn-point","seed":202610077026,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[\"origin\"],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(0, 0) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (0, 0) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0027","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90bd4ddeafbcae6e6d39ac8824f06fffe2dbfce591969d1649217c0320511e77","family_id":"qf-transform-quarter-turn-point","seed":202610077027,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-11\",\"2\"],1],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-11, 2) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-2, -11)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-2, -11) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0028","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98a134ca584bbe0af13388ac20998dd0fbee4ecc9e0cc5ce5c68f1deb5ecd785","family_id":"qf-transform-quarter-turn-point","seed":202610077028,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-10\",\"3\"],2],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-10, 3) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(10, -3)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (10, -3) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0029","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba867a2cba19c4638ceb52908c8b01f94bca0741a864b9bee2cff4238245a160","family_id":"qf-transform-quarter-turn-point","seed":202610077029,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-9\",\"4\"],3],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-9, 4) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(4, 9)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (4, 9) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0030","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7351eedc76bf02d6f067250358b547f9e318b57df83dcf7131e11bbef07ef400","family_id":"qf-transform-quarter-turn-point","seed":202610077030,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-8\",\"5\"],0],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-8, 5) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-8, 5)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-8, 5) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0031","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc37c7eb61754082ecc4d854fb742e9fbc4ae66a6839d926f789da025299b55a","family_id":"qf-transform-quarter-turn-point","seed":202610077031,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-7\",\"6\"],1],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-7, 6) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-6, -7)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-6, -7) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0032","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"689ef981ac7db93087547d83bcbb8feb903521398e18f511d570778f0b852cee","family_id":"qf-transform-quarter-turn-point","seed":202610077032,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-6\",\"7\"],2],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-6, 7) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(6, -7)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (6, -7) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0033","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49045d4423ce30888ba7a60d08d1f0dc85e061a602dcb6d8d6e8bc1423db39d3","family_id":"qf-transform-quarter-turn-point","seed":202610077033,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-5\",\"8\"],3],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-5, 8) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(8, 5)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (8, 5) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0034","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ebb16be1066fe6949bbd87f23479d5c0cdb138ab46e80b3cf24d26db3c09fd3","family_id":"qf-transform-quarter-turn-point","seed":202610077034,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-4\",\"9\"],0],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-4, 9) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-4, 9)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-4, 9) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0035","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f07d14c31cb3bf5c10b7d88f762aa02aec34c836e67659c2d4cb2a342a7c03e","family_id":"qf-transform-quarter-turn-point","seed":202610077035,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-3\",\"10\"],1],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-3, 10) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-10, -3)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-10, -3) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0036","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e245ba7769ba291766abe48abc4b494920789f9fc34aad67ebbb3956e8e3dd30","family_id":"qf-transform-quarter-turn-point","seed":202610077036,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-2\",\"11\"],2],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-2, 11) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(2, -11)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (2, -11) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0037","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b67f947919a5db0138c444422a9f504e59bd8ed0117ec84ad1ea766a34bb4ad9","family_id":"qf-transform-quarter-turn-point","seed":202610077037,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-1\",\"12\"],3],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-1, 12) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(12, 1)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (12, 1) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0038","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6b350c5036085fbcdb52bbd48fb36655b6b3371e6f0e89b9228a4ac4ec6d1fc","family_id":"qf-transform-quarter-turn-point","seed":202610077038,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"0\",\"13\"],0],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(0, 13) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(0, 13)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (0, 13) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0039","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64b30d0dc077ff3554c9577d08321d1504e7619b7a80bd7a763609d0cd685307","family_id":"qf-transform-quarter-turn-point","seed":202610077039,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"1\",\"14\"],1],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(1, 14) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-14, 1)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-14, 1) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0040","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc2643953d7b3c592ec92c23c05ee407a578802fbcf1cc50de359d5d015f4eb2","family_id":"qf-transform-quarter-turn-point","seed":202610077040,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"2\",\"15\"],2],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(2, 15) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-2, -15)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-2, -15) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0041","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8180e46eee4b87f53791dc3f6a8103a858ee40d5edab7f1e6f23cd5860274d5e","family_id":"qf-transform-quarter-turn-point","seed":202610077041,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"3\",\"16\"],3],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(3, 16) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(16, -3)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (16, -3) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0042","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aec632a28d2f441d83de5cacbcad19c189d874026cfd7245a5ec3c5a8de211e0","family_id":"qf-transform-quarter-turn-point","seed":202610077042,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"4\",\"17\"],0],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(4, 17) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(4, 17)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (4, 17) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0043","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2446fd01846d764aa44f02b83da289b24bd840932857ea06c600c2d3a00814a7","family_id":"qf-transform-quarter-turn-point","seed":202610077043,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"5\",\"18\"],1],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(5, 18) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-18, 5)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-18, 5) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0044","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"557929b4cd99c455556a117b9787d7f7f564f13563c5cfafb26fcc4e5b3f54d2","family_id":"qf-transform-quarter-turn-point","seed":202610077044,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"6\",\"19\"],2],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(6, 19) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-6, -19)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-6, -19) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0045","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79d3f1c030c164a15258d4e5b12f2f8fb0d39be218da012a033c9d146540a330","family_id":"qf-transform-quarter-turn-point","seed":202610077045,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"7\",\"20\"],3],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(7, 20) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(20, -7)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (20, -7) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0046","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc859b60326c422bd2d2a557b1f1fec1bd03ed198d184466da6ff8f865e7ec41","family_id":"qf-transform-quarter-turn-point","seed":202610077046,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"8\",\"21\"],0],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(8, 21) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(8, 21)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (8, 21) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0047","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"110e14513396bcf93bceba98f2cdb8e3e9f4ec104309dbf1cbc7f771714fc463","family_id":"qf-transform-quarter-turn-point","seed":202610077047,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"9\",\"22\"],1],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(9, 22) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-22, 9)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-22, 9) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0048","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdec88681c6e8b99cb579f215f6a7059016c5da894c1bfa5496da35eb0ebffc5","family_id":"qf-transform-quarter-turn-point","seed":202610077048,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"10\",\"23\"],2],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(10, 23) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(-10, -23)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (-10, -23) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0049","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4896570e4c20331bf6745f0b2c4c4d85fcb6e46a8ab609600fd80c20f1a9bf1","family_id":"qf-transform-quarter-turn-point","seed":202610077049,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"11\",\"24\"],3],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(11, 24) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(24, -11)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (24, -11) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0050","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dbd500e528b19713b4c0a5f65858f147f3b37b1b0a7a87da95cdd5160c49195","family_id":"qf-transform-quarter-turn-point","seed":202610077050,"semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"12\",\"25\"],0],\"operation\":\"quarter_turn_point\"}"},"fields":{"title":"Counterclockwise Quarter-Turns about the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(12, 25) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.","answer":{"kind":"expression","value":"(12, 25)","explanation":"Each quarter-turn sends (x,y) to (−y,x), giving (12, 25) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property."},"skills":["Counterclockwise Quarter-Turns about the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0051","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b98863fde3d843b4cf68af7e2aa32446323acecad6f9a8d26f0a42a7bd709a2a","family_id":"qf-transform-axis-reflection-point","seed":202610077051,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[\"origin\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(0, 0) across the x-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Use the specified reflection formula (x,y)→(x,−y), giving (0, 0). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0052","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d321220147f08d1fe72c0a40ee7a8656f3aeba991cd4037f70f9653920c6e14f","family_id":"qf-transform-axis-reflection-point","seed":202610077052,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-11\",\"3\"],\"y\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-11, 3) across the y-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(11, 3)","explanation":"Use the specified reflection formula (x,y)→(−x,y), giving (11, 3). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0053","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc615f54c8589b4b511fe0898aaf355254c44a65bc9465ef9fd2ad42a1e70e0f","family_id":"qf-transform-axis-reflection-point","seed":202610077053,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-10\",\"4\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-10, 4) across the line y=x. Find the image coordinates.","answer":{"kind":"expression","value":"(4, -10)","explanation":"Use the specified reflection formula (x,y)→(y,x), giving (4, -10). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0054","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9642da3903e86b437e460ff6b5282c2c015b10b66666049b7b1b31acb278ffb5","family_id":"qf-transform-axis-reflection-point","seed":202610077054,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-9\",\"5\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-9, 5) across the line y=−x. Find the image coordinates.","answer":{"kind":"expression","value":"(-5, 9)","explanation":"Use the specified reflection formula (x,y)→(−y,−x), giving (-5, 9). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0055","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9b1c167fb0911b4fc7577dd51a340e3414cb51b3653dccf249d1a0d921a9174","family_id":"qf-transform-axis-reflection-point","seed":202610077055,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-8\",\"6\"],\"x\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-8, 6) across the x-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(-8, -6)","explanation":"Use the specified reflection formula (x,y)→(x,−y), giving (-8, -6). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0056","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7e194a0bdf35c1133ed1fb5ac7f15c0aa0e3118f13a1458f77ea6703a644e09","family_id":"qf-transform-axis-reflection-point","seed":202610077056,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-7\",\"7\"],\"y\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-7, 7) across the y-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(7, 7)","explanation":"Use the specified reflection formula (x,y)→(−x,y), giving (7, 7). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0057","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"789ad394af8e565974e3f1149d14f1b68de50df315fada84c5dda363936290e1","family_id":"qf-transform-axis-reflection-point","seed":202610077057,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-6\",\"8\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-6, 8) across the line y=x. Find the image coordinates.","answer":{"kind":"expression","value":"(8, -6)","explanation":"Use the specified reflection formula (x,y)→(y,x), giving (8, -6). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0058","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec5b596d7b014e966716ed9093b69949f5b1f78f252650b1dd0b494a40487ed","family_id":"qf-transform-axis-reflection-point","seed":202610077058,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-5\",\"9\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-5, 9) across the line y=−x. Find the image coordinates.","answer":{"kind":"expression","value":"(-9, 5)","explanation":"Use the specified reflection formula (x,y)→(−y,−x), giving (-9, 5). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0059","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afff489dee78c8728ff911b98f3884e1b9ef40d138f4129eef43fee4e49716c9","family_id":"qf-transform-axis-reflection-point","seed":202610077059,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-4\",\"10\"],\"x\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-4, 10) across the x-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(-4, -10)","explanation":"Use the specified reflection formula (x,y)→(x,−y), giving (-4, -10). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0060","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3f121d392aca32656b345f4ac6d4cbae523651723577fe9cf7894a8a9634eb0","family_id":"qf-transform-axis-reflection-point","seed":202610077060,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-3\",\"11\"],\"y\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-3, 11) across the y-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(3, 11)","explanation":"Use the specified reflection formula (x,y)→(−x,y), giving (3, 11). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0061","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0377cf4a26229259df035f15a7e21e5d9bf132c600f5fdc46345853fdf5ce21","family_id":"qf-transform-axis-reflection-point","seed":202610077061,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-2\",\"12\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-2, 12) across the line y=x. Find the image coordinates.","answer":{"kind":"expression","value":"(12, -2)","explanation":"Use the specified reflection formula (x,y)→(y,x), giving (12, -2). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0062","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c4733e886e9499b9ed6bca9bc6ab42504545d8d0a03dd7dec5d470c66cc7056","family_id":"qf-transform-axis-reflection-point","seed":202610077062,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-1\",\"13\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-1, 13) across the line y=−x. Find the image coordinates.","answer":{"kind":"expression","value":"(-13, 1)","explanation":"Use the specified reflection formula (x,y)→(−y,−x), giving (-13, 1). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0063","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"184dde6a4e012f1f523ccf87d73ede0683ae1b0e3e745c77f58a4fd820e39a10","family_id":"qf-transform-axis-reflection-point","seed":202610077063,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"0\",\"14\"],\"x\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(0, 14) across the x-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(0, -14)","explanation":"Use the specified reflection formula (x,y)→(x,−y), giving (0, -14). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0064","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dd3fa99bffd6ca6305174d89882d64881ee9afe7413be38e3798c3aa2d140d5","family_id":"qf-transform-axis-reflection-point","seed":202610077064,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"1\",\"15\"],\"y\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(1, 15) across the y-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(-1, 15)","explanation":"Use the specified reflection formula (x,y)→(−x,y), giving (-1, 15). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0065","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62cc34a5c4a6693553cd469816033d7408b28afa07e2d650c04c8948f21030d3","family_id":"qf-transform-axis-reflection-point","seed":202610077065,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"2\",\"16\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(2, 16) across the line y=x. Find the image coordinates.","answer":{"kind":"expression","value":"(16, 2)","explanation":"Use the specified reflection formula (x,y)→(y,x), giving (16, 2). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0066","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2d7128795da01ed0612ae748efdfb147eab10196e8b91b823f032d1afef63f3","family_id":"qf-transform-axis-reflection-point","seed":202610077066,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"3\",\"17\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(3, 17) across the line y=−x. Find the image coordinates.","answer":{"kind":"expression","value":"(-17, -3)","explanation":"Use the specified reflection formula (x,y)→(−y,−x), giving (-17, -3). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0067","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"531cb69e0540f7547ac5968d79437fb9c95cfba3a71fba37ff4970465f11ed8a","family_id":"qf-transform-axis-reflection-point","seed":202610077067,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"4\",\"18\"],\"x\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(4, 18) across the x-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(4, -18)","explanation":"Use the specified reflection formula (x,y)→(x,−y), giving (4, -18). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0068","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1831fb5d9e8920ede677423bdc488e1cd3aaf40fc6d1a9606d1fcc93638e8ee6","family_id":"qf-transform-axis-reflection-point","seed":202610077068,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"5\",\"19\"],\"y\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(5, 19) across the y-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(-5, 19)","explanation":"Use the specified reflection formula (x,y)→(−x,y), giving (-5, 19). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0069","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96c2dbad9f12ba35aed910e08e5cba7b73e2477fe12c6c037b59a69baea4e155","family_id":"qf-transform-axis-reflection-point","seed":202610077069,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"6\",\"20\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(6, 20) across the line y=x. Find the image coordinates.","answer":{"kind":"expression","value":"(20, 6)","explanation":"Use the specified reflection formula (x,y)→(y,x), giving (20, 6). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0070","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e829293b9060f29862908d1e6ed3c508124c922a811931dcc86e65fff7aa16a6","family_id":"qf-transform-axis-reflection-point","seed":202610077070,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"7\",\"21\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(7, 21) across the line y=−x. Find the image coordinates.","answer":{"kind":"expression","value":"(-21, -7)","explanation":"Use the specified reflection formula (x,y)→(−y,−x), giving (-21, -7). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0071","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a09d0c8af6684c4b6460530847ec7ae0e4d6655dc88eec71a61055218f98d06","family_id":"qf-transform-axis-reflection-point","seed":202610077071,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"8\",\"22\"],\"x\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(8, 22) across the x-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(8, -22)","explanation":"Use the specified reflection formula (x,y)→(x,−y), giving (8, -22). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0072","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2272e955475c98972560690d17a773baac11b1e4f90d7d1e1da604343653942","family_id":"qf-transform-axis-reflection-point","seed":202610077072,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"9\",\"23\"],\"y\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(9, 23) across the y-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(-9, 23)","explanation":"Use the specified reflection formula (x,y)→(−x,y), giving (-9, 23). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0073","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5714913fd907a25c38bc22e3f39f27c360e4ad31196017238ae06c4aee739f56","family_id":"qf-transform-axis-reflection-point","seed":202610077073,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"10\",\"24\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(10, 24) across the line y=x. Find the image coordinates.","answer":{"kind":"expression","value":"(24, 10)","explanation":"Use the specified reflection formula (x,y)→(y,x), giving (24, 10). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0074","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ca882015a9aa671bca6a30c1a405e64c347857eab806b85db5664378c75f46f","family_id":"qf-transform-axis-reflection-point","seed":202610077074,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"11\",\"25\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(11, 25) across the line y=−x. Find the image coordinates.","answer":{"kind":"expression","value":"(-25, -11)","explanation":"Use the specified reflection formula (x,y)→(−y,−x), giving (-25, -11). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0075","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bffe96b654c52cb4b7b61e25f570201b5a66d043d57aafa81d69b6fb063874a0","family_id":"qf-transform-axis-reflection-point","seed":202610077075,"semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"12\",\"26\"],\"x\"],\"operation\":\"axis_reflection_point\"}"},"fields":{"title":"Reflection across a Specified Line through the Origin","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(12, 26) across the x-axis. Find the image coordinates.","answer":{"kind":"expression","value":"(12, -26)","explanation":"Use the specified reflection formula (x,y)→(x,−y), giving (12, -26). A line reflection is not central inversion (−x,−y)."},"skills":["Reflection across a Specified Line through the Origin"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0076","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e93cb743d630690d58128d1aefb6888cbfb810637549202dba157f733347e2fd","family_id":"qf-transform-uniform-scale-class","seed":202610077076,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"0\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(0)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"degenerate constant map","explanation":"Classify the global formula by |c|, giving degenerate constant map: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0077","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37784bdb11ceaac4acc2dcb330164731ce5445c830f9327423fabc14ad391fe1","family_id":"qf-transform-uniform-scale-class","seed":202610077077,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"1\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-1)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"whole-plane isometry","explanation":"Classify the global formula by |c|, giving whole-plane isometry: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0078","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5489bd81404b84331666d95f3f235b8d828a41589f864ce9105c05f43b44660","family_id":"qf-transform-uniform-scale-class","seed":202610077078,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"2\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(2)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=2","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=2: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0079","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dbee65fe5b2d1f11db9a6811ae1b6dd92fb9eba01cde57085fc12f9735bd3a2","family_id":"qf-transform-uniform-scale-class","seed":202610077079,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"3\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-3)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=3","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=3: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0080","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a60b6b87074c34865236022657ee27048e8b8f09fadc1ee2c08dd51312c22f4","family_id":"qf-transform-uniform-scale-class","seed":202610077080,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"4\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(4)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=4","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=4: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0081","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b06ef3a2598ca389dbd0fb145119d62fa2e9a453213739918c98f85f84d15ab1","family_id":"qf-transform-uniform-scale-class","seed":202610077081,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"5\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-5)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=5","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=5: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0082","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5237559c4e78f498c8fea781835f38d44c257329ef164cfd8020af441ea7af8e","family_id":"qf-transform-uniform-scale-class","seed":202610077082,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"6\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(6)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=6","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=6: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0083","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74fa1e857e53c1dd60bd5018e4c9fb47b76d9866818db3ca793a27d5624c3fa7","family_id":"qf-transform-uniform-scale-class","seed":202610077083,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"7\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-7)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=7","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=7: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0084","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce59546e337e6a855671224233ccf9e72f39e9a0020f695f80ae3a5c93032ea2","family_id":"qf-transform-uniform-scale-class","seed":202610077084,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"8\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(8)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=8","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=8: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0085","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b015923aa96a21dff92b48e964488c69e11a15900f2d275fa6a2eed3c8b0fd3","family_id":"qf-transform-uniform-scale-class","seed":202610077085,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"9\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-9)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=9","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=9: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0086","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5d861cd9179d0fc8d65dde8d07420e57a4bf51d38a3b049fd290a9686bfe8b3","family_id":"qf-transform-uniform-scale-class","seed":202610077086,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"10\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(10)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=10","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=10: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0087","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47bd8494c9b55294a49abe7fb061def5d72a078630b210b78274b8b6f7629546","family_id":"qf-transform-uniform-scale-class","seed":202610077087,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"11\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-11)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=11","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=11: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0088","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ef3fc8846b2c05df11b5745b5db6e41daf8b5ddb786f3fb31ca2b67cb215e6","family_id":"qf-transform-uniform-scale-class","seed":202610077088,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"12\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(12)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=12","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=12: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0089","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"621f921bf15608df8ec1982a4b30e3c3cfdd0f8bdf6ed722ae2bb8071dad13e1","family_id":"qf-transform-uniform-scale-class","seed":202610077089,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"13\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-13)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=13","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=13: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0090","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ad591fa53321337ffd6b048d19cc5c654234eee1bd21858d9b9347678069e4f","family_id":"qf-transform-uniform-scale-class","seed":202610077090,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"14\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(14)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=14","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=14: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0091","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fed2559e03f0f5de4d39c6578582d4691ad24f58be589e7740713d75eaf7ae5","family_id":"qf-transform-uniform-scale-class","seed":202610077091,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"15\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-15)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=15","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=15: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0092","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"062ac420edc597cf1448d7dcb9cf5e34bb07c45c696a2a15ce50870533bad5e7","family_id":"qf-transform-uniform-scale-class","seed":202610077092,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"16\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(16)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=16","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=16: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0093","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9883366c59abfe44a36dc80b5ad171a32932cda00a172cd85bb6c7710ec19d50","family_id":"qf-transform-uniform-scale-class","seed":202610077093,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"17\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-17)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=17","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=17: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0094","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5db6980ea8603e435f456e0bea8456cee907fd630b247db19f50694ce319a23e","family_id":"qf-transform-uniform-scale-class","seed":202610077094,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"18\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(18)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=18","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=18: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0095","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c902afc958c4dbdc786ce12c1454589c59d3450e2752a02c6dc8bab27e476ca4","family_id":"qf-transform-uniform-scale-class","seed":202610077095,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"19\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-19)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=19","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=19: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0096","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d31157fae5a66f63d7d2a39007f54cbad67e18bcd4a7f83c2f9b8aa223ea40","family_id":"qf-transform-uniform-scale-class","seed":202610077096,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"20\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(20)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=20","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=20: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0097","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e2841785e732314a3c93d4b9caf26f9ada3fbf176fd9521719f1b77d259e049","family_id":"qf-transform-uniform-scale-class","seed":202610077097,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"21\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-21)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=21","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=21: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0098","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91f6ac415654ec41e516f0a0de736ba79419fd7c62d27eda30bdcca7ddcbca13","family_id":"qf-transform-uniform-scale-class","seed":202610077098,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"22\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(22)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=22","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=22: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0099","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7ece709a64712ff668a4776f831beffbba78a913fe662a045200233caf07a98","family_id":"qf-transform-uniform-scale-class","seed":202610077099,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"23\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-23)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=23","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=23: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r07-geometric-transformations-0100","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d01bab4aa3b075d86d9ddebd2acb3fa1d0842bf41206d9dad52898b807683489","family_id":"qf-transform-uniform-scale-class","seed":202610077100,"semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"24\"],\"operation\":\"uniform_scale_class\"}"},"fields":{"title":"Whole-Plane Classification of Uniform Scaling","stem":"Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(24)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.","answer":{"kind":"short_text","value":"non-isometric similarity; scale=24","explanation":"Classify the global formula by |c|, giving non-isometric similarity; scale=24: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry."},"skills":["Whole-Plane Classification of Uniform Scaling"],"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."}}