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.
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.
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.
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.
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.
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.
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.
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.
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.