# Question 34506: Missing Side by the Pythagorean Theorem [Return to catalogue page 173](https://ocme.evemisslab.com/en/questions/page/173/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260912-geometry-foundations/#q-20260912-pythagorean-missing-side-0006) 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. R0068-34506 · q-20260912-pythagorean-missing-side-0006 ## Original question and givens A right triangle has one leg of length 40 and a hypotenuse of length 41. Find the other leg length a. - G001 — A right triangle has one leg of length 40 and a hypotenuse of length 41. Find the other leg length a. ## 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-euclid-pythagorean-theorem](https://ocme.evemisslab.com/en/mko/mko-euclid-pythagorean-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-pythagorean-missing-side, [hypotenuse, known, legs, mode]] After derived = [BASE-no-required-operation, no-invented-domain] Conditions: none Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact. Step identifiers and dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001 S002 · impliesBefore source roles = [leg, 40, 41] After derived = [right triangle premise, unknown a>0, a^2=c^2-b^2] Conditions: none The ORIGINAL bilingual known list is[KNOWN leg,hypotenuse],and source legs[1]is THAT known leg.Use ONLY these givens,NEVER the unknown-filled legs[0]as an answer.FIRST require a positive known leg shorter than the hypotenuse. Step identifiers and dependenciesRule: Pythagorean_GIVEN_known_leg_hypotenuse_NOT_unknown_filled_leg · Uses: G001, S001 S003 · calculationBefore Pythagorean ORIGINAL given hypotenuse squared := mul(41, 41) After Pythagorean ORIGINAL given hypotenuse squared = 1681 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Pythagorean ORIGINAL given known leg squared := mul(40, 40) After Pythagorean ORIGINAL given known leg squared = 1600 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Pythagorean ORIGINAL hypotenuse square minus known leg square := sub(1681, 1600) After Pythagorean ORIGINAL hypotenuse square minus known leg square = 81 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Pythagorean DERIVED positive root squared for reverse substitution := mul(9, 9) After Pythagorean DERIVED positive root squared for reverse substitution = 81 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [missing leg, 81] After derived = [9, 81] Conditions: none The unknown side has square radicand;substitute back to verify value² exactly.A POSITIVE length selects the positive principal root,not BOTH±branches.All25 ACTUAL roots are exact integers;do not use Math.sqrt approximations or copy filled answer metadata. Step identifiers and dependenciesRule: Pythagorean_POSITIVE_length_selects_principal_exact_root · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S006 · premise_and_result_audit — Recompute Pythagorean DERIVED positive root squared for reverse substitution with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Pythagorean ORIGINAL hypotenuse square minus known leg square with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Pythagorean ORIGINAL given known leg squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Pythagorean ORIGINAL given hypotenuse squared with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. ## Result 9 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: 3837df51c2cc81f0538e6b437536633c6fb32410f3f41f6dd8b715f6c7ad93a5 - candidate record SHA-256: a13206c2d81511d5ba810f8db3f5b93d4e57558e25493f9f2a0a8af2ce7af59f - semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed - question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260912-pythagorean-missing-side-0006.json)