Question 31385: Nonnegative Distance in a Perfect-Square Slice

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-123, 18). 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=(-3, -4) and Q=(-123, 18). 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(-123, -3)

Distance PQ ORIGINAL Q minus P axis0 = -120

Distance PQ ORIGINAL Q minus P axis1 := sub(18, -4)

Distance PQ ORIGINAL Q minus P axis1 = 22

Distance PQ ORIGINAL signed delta squared axis0 := mul(-120, -120)

Distance PQ ORIGINAL signed delta squared axis0 = 14400

Distance PQ ORIGINAL signed delta squared axis1 := mul(22, 22)

Distance PQ ORIGINAL signed delta squared axis1 = 484

Distance PQ ORIGINAL sum of two squares := sum(14400, 484)

Distance PQ ORIGINAL sum of two squares = 14884

source roles = [Distance PQ, [-3, -4], [-123, 18]]

derived = [[-120, 22], [14400, 484], 14884]

Distance reverse principal root square := mul(122, 122)

Distance reverse principal root square = 14884

source roles = [14884]

derived = [122, 14884, 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

122

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.