Question 31389: Nonnegative Distance in a Perfect-Square Slice

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-223, 22). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-223, 22). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.

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-distance-formula

Step-by-step derivation

source roles = [qf-distance-perfect-square, perfect_square_distance, frame, euclidean_orthonormal_equal_units]

derived = [[integer_principal_root]]

Distance PQ ORIGINAL Q minus P axis0 := sub(-223, 1)

Distance PQ ORIGINAL Q minus P axis0 = -224

Distance PQ ORIGINAL Q minus P axis1 := sub(22, -8)

Distance PQ ORIGINAL Q minus P axis1 = 30

Distance PQ ORIGINAL signed delta squared axis0 := mul(-224, -224)

Distance PQ ORIGINAL signed delta squared axis0 = 50176

Distance PQ ORIGINAL signed delta squared axis1 := mul(30, 30)

Distance PQ ORIGINAL signed delta squared axis1 = 900

Distance PQ ORIGINAL sum of two squares := sum(50176, 900)

Distance PQ ORIGINAL sum of two squares = 51076

source roles = [Distance PQ, [1, -8], [-223, 22]]

derived = [[-224, 30], [50176, 900], 51076]

Distance reverse principal root square := mul(226, 226)

Distance reverse principal root square = 51076

source roles = [51076]

derived = [226, 51076, Yes]

Reverse checks

• S009 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S008 · premise_and_result_audit — Recompute Distance reverse principal root square with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S006 · premise_and_result_audit — Recompute Distance PQ ORIGINAL sum of two squares with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Distance PQ ORIGINAL signed delta squared axis1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Distance PQ ORIGINAL signed delta squared axis0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Distance PQ ORIGINAL Q minus P axis1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Distance PQ ORIGINAL Q minus P axis0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.

Result

226

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.