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R0011-06391 · q-20261009-advance-20261013-r01-pythagorean-theorem-0066
Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=48, positive hypotenuse c=80>a; c²−a² is a positive perfect square. Find the other positive leg b.
G001 — Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=48, positive hypotenuse c=80>a; c²−a² is a positive perfect square. Find the other positive leg b.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.
Before
Euclidean right-side model;positive leg a=48
After
use the supplied right-angle premise:a²+b²=c²
Conditions: none
Use the declared Euclidean right-side model with positive legs.This finite numerical check is not a new universal proof of the Pythagorean theorem or an arbitrary affine-geometric bridge.
Rule: pythagorean_right_side_premise · Uses: G001
Before
known leg a squared := square(48)
After
known leg a squared = 2304
Conditions: none
Square the complete known positive leg a.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
hypotenuse c squared := square(80)
After
hypotenuse c squared = 6400
Conditions: none
Square the positive hypotenuse c.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
missing leg squared := sub(6400, 2304)
After
missing leg squared = 4096
Conditions: none
Rearrange b²=c²−a² by subtracting the known leg square;the result must be a positive perfect square.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
missing leg principal root squared := square(64)
After
missing leg principal root squared = 4096
Conditions: none
Square the integer root to audit the positive-perfect-square premise,without rounded floating roots.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
root=64>0;root²=4096=required square
After
unique positive length=64
Conditions: none
Matching the square still requires a positive length;a negative root is not a side length,and0 violates this nondegenerate positive-leg protocol.
Rule: positive_integer_length_root · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute missing leg principal root squared with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute missing leg squared with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute hypotenuse c squared with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute known leg a squared with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.64
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 040851b8c367b176c3480202c248d458a4aae5b5e05d1c9441d65b73033849f3candidate record SHA-256: 50fc5328db05a3fca3a461d064977cc4eb09822fd1493d854357ef054f11b268semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED