# Question 28848: Conditional Scaling of a Given Positive Area [Return to catalogue page 145](https://ocme.evemisslab.com/en/questions/page/145/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r06-similarity-foundations/#q-20261006-advance-20261007-r06-similarity-0098) 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. R0056-28848 · q-20261006-advance-20261007-r06-similarity-0098 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 45/2>0 squared units. At corresponding positive length scale k=3/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers. - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 45/2>0 squared units. At corresponding positive length scale k=3/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers. ## 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-similarity](https://ocme.evemisslab.com/en/mko/mko-similarity/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-similarity-scaled-area, scaled_area] After derived = [given_positive_area_times_squared_scale] Conditions: none Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions. Step identifiers and dependenciesRule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001 S002 · calculationBefore Similarity ORIGINAL scale numerator squared := mul(3, 3) After Similarity ORIGINAL scale numerator squared = 9 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Similarity ORIGINAL scale denominator squared := mul(2, 2) After Similarity ORIGINAL scale denominator squared = 4 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Similarity ORIGINAL given area numerator times p squared := mul(45, 9) After Similarity ORIGINAL given area numerator times p squared = 405 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Similarity ORIGINAL given area denominator times q squared := mul(2, 4) After Similarity ORIGINAL given area denominator times q squared = 8 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Similarity ORIGINAL scaled area fraction := div(405, 8) After Similarity ORIGINAL scaled area fraction = 405/8 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [scaled_area, [45, 2, 3, 2]] After derived = [9, 4, 405/8] Conditions: none Lengths scale by k=p/q,so squared lengths and area scale by k²,not k.Two independent length directions give determinant factor k².This is conditional conversion of the GIVEN positive area,not a coordinate-area recomputation or a general measure theorem. Step identifiers and dependenciesRule: Similarity_GIVEN_positive_square_length_or_area_times_SCALE_SQUARED · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S006 · premise_and_result_audit — Recompute Similarity ORIGINAL scaled area fraction with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Similarity ORIGINAL given area denominator times q squared with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Similarity ORIGINAL given area numerator times p squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Similarity ORIGINAL scale denominator squared with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Similarity ORIGINAL scale numerator squared with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. ## Result 405/8 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: 45e9088a1315f2699f85844e6e48335caee676c35ec3d53dd5e2cc04c796dc03 - candidate record SHA-256: 64c41e4c58c0039806c8b2782057611fc095c37c1957c99049c31123e9aca6d3 - semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8 - question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r06-similarity-0098.json)