# Question 35545: Coordinate-Plane Distance [Return to catalogue page 178](https://ocme.evemisslab.com/en/questions/page/178/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260912-geometry-foundations/#q-20260912-distance-coordinate-0030) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0070-35545 · q-20260912-distance-coordinate-0030 ## Original question and givens In the Euclidean plane, P=(-8, -25) and Q=(-74, 87). Find the distance d(P,Q). - G001 — In the Euclidean plane, P=(-8, -25) and Q=(-74, 87). Find the distance d(P,Q). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-euclidean-length](https://ocme.evemisslab.com/en/mko/mko-euclidean-length/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-distance-coordinate, [expected_distance, p, q]] After derived = [distance, null, null] Conditions: none Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules. Step identifiers and dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001 S002 · impliesBefore source roles = [[-8, -25], [-74, 87]] After derived = [orthonormal common coordinate frame] Conditions: none These LAST FIVE questions give integral coordinates in ONE Euclidean frame.Distance is neither the sum of coordinate differences nor Manhattan distance;expected_distance metadata is NOT an oracle. Step identifiers and dependenciesRule: Distance_GIVEN_same_Euclidean_frame_integral_last5 · Uses: G001, S001 S003 · calculationBefore Distance ORIGINAL coordinate difference i=0 := sub(-74, -8) After Distance ORIGINAL coordinate difference i=0 = -66 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Distance square signed difference i=0 := mul(-66, -66) After Distance square signed difference i=0 = 4356 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Distance ORIGINAL coordinate difference i=1 := sub(87, -25) After Distance ORIGINAL coordinate difference i=1 = 112 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Distance square signed difference i=1 := mul(112, 112) After Distance square signed difference i=1 = 12544 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Distance sum BOTH coordinate squares := sum(4356, 12544) After Distance sum BOTH coordinate squares = 16900 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Distance principal nonnegative root squared back := mul(130, 130) After Distance principal nonnegative root squared back = 16900 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore source roles = [[-66, 112], 16900] After derived = [130] Conditions: none The sum of BOTH squared components is d².Distance is nonnegative,so choose the principal root and square it back EXACTLY.Neither a negative root nor expected-answer backsolving is allowed. Step identifiers and dependenciesRule: Distance_PRINCIPAL_root_not_negative_or_approximate · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S008 · premise_and_result_audit — Recompute Distance principal nonnegative root squared back with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Distance sum BOTH coordinate squares with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Distance square signed difference i=1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Distance ORIGINAL coordinate difference i=1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Distance square signed difference i=0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Distance ORIGINAL coordinate difference i=0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. ## Result 130 The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 07c0416627cbc34bc595ae6911139695a17bade3aec3dc3a88404c03cc9646bb - candidate record SHA-256: f9f768e936724f3f221899c97cde7fc5f700ab0617e7a73130af024d4ad6f740 - semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80 - question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260912-distance-coordinate-0030.json)