Question 31350: Sign and Square Eligibility of a Candidate Distance

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 6). Is the candidate d=25 the Euclidean L2 distance? Check both its sign and its square.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 6). Is the candidate d=25 the Euclidean L2 distance? Check both its sign and its square.

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-candidate, candidate_distance, frame, euclidean_orthonormal_equal_units]

derived = [[nonnegative_exact_candidate_YesNo]]

Distance PQ ORIGINAL Q minus P axis0 := sub(12, 12)

Distance PQ ORIGINAL Q minus P axis0 = 0

Distance PQ ORIGINAL Q minus P axis1 := sub(6, -19)

Distance PQ ORIGINAL Q minus P axis1 = 25

Distance PQ ORIGINAL signed delta squared axis0 := mul(0, 0)

Distance PQ ORIGINAL signed delta squared axis0 = 0

Distance PQ ORIGINAL signed delta squared axis1 := mul(25, 25)

Distance PQ ORIGINAL signed delta squared axis1 = 625

Distance PQ ORIGINAL sum of two squares := sum(0, 625)

Distance PQ ORIGINAL sum of two squares = 625

source roles = [Distance PQ, [12, -19], [12, 6]]

derived = [[0, 25], [0, 625], 625]

Distance ORIGINAL candidate d squared := mul(25, 25)

Distance ORIGINAL candidate d squared = 625

source roles = [25, 625, 625]

derived = [Yes, Yes, 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 ORIGINAL candidate d squared 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

Yes

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.