# Question 6386: Positive Integer Missing Leg

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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.

R0011-06386 · q-20261009-advance-20261013-r01-pythagorean-theorem-0061



## Original question and givens

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=33, positive hypotenuse c=55>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=33, positive hypotenuse c=55>a; c²−a² is a positive perfect square. Find the other positive leg b.





## 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

Euclidean right-side model;positive leg a=33

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.

Step identifiers and dependenciesRule: pythagorean_right_side_premise · Uses: G001





S002 · calculationBefore

known leg a squared := square(33)

After

known leg a squared = 1089

Conditions: none

Square the complete known positive leg a.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

hypotenuse c squared := square(55)

After

hypotenuse c squared = 3025

Conditions: none

Square the positive hypotenuse c.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

missing leg squared := sub(3025, 1089)

After

missing leg squared = 1936

Conditions: none

Rearrange b²=c²−a² by subtracting the known leg square;the result must be a positive perfect square.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

missing leg principal root squared := square(44)

After

missing leg principal root squared = 1936

Conditions: none

Square the integer root to audit the positive-perfect-square premise,without rounded floating roots.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

root=44>0;root²=1936=required square

After

unique positive length=44

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.

Step identifiers and dependenciesRule: positive_integer_length_root · Uses: G001, S005









## Reverse checks


- 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.





## Result

44

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: b8977d987826df7bce7c8a6c7704e09e964bb3f8f00ae4533e0944f28fe86b01
- candidate record SHA-256: 951806242995854f0ae21ddc2fdbdadb81ccc8b13a710f9594eaf903da067141
- semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
- question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0061.json)