Question 31381: Nonnegative Distance in a Perfect-Square Slice

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-55, 14). 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=(-7, 0) and Q=(-55, 14). 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(-55, -7)

Distance PQ ORIGINAL Q minus P axis0 = -48

Distance PQ ORIGINAL Q minus P axis1 := sub(14, 0)

Distance PQ ORIGINAL Q minus P axis1 = 14

Distance PQ ORIGINAL signed delta squared axis0 := mul(-48, -48)

Distance PQ ORIGINAL signed delta squared axis0 = 2304

Distance PQ ORIGINAL signed delta squared axis1 := mul(14, 14)

Distance PQ ORIGINAL signed delta squared axis1 = 196

Distance PQ ORIGINAL sum of two squares := sum(2304, 196)

Distance PQ ORIGINAL sum of two squares = 2500

source roles = [Distance PQ, [-7, 0], [-55, 14]]

derived = [[-48, 14], [2304, 196], 2500]

Distance reverse principal root square := mul(50, 50)

Distance reverse principal root square = 2500

source roles = [2500]

derived = [50, 2500, 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

50

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.