Question 31341: Sign and Square Eligibility of a Candidate Distance

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-285, 24). Is the candidate d=290 the Euclidean L2 distance? Check both its sign and its square.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-285, 24). Is the candidate d=290 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(-285, 3)

Distance PQ ORIGINAL Q minus P axis0 = -288

Distance PQ ORIGINAL Q minus P axis1 := sub(24, -10)

Distance PQ ORIGINAL Q minus P axis1 = 34

Distance PQ ORIGINAL signed delta squared axis0 := mul(-288, -288)

Distance PQ ORIGINAL signed delta squared axis0 = 82944

Distance PQ ORIGINAL signed delta squared axis1 := mul(34, 34)

Distance PQ ORIGINAL signed delta squared axis1 = 1156

Distance PQ ORIGINAL sum of two squares := sum(82944, 1156)

Distance PQ ORIGINAL sum of two squares = 84100

source roles = [Distance PQ, [3, -10], [-285, 24]]

derived = [[-288, 34], [82944, 1156], 84100]

Distance ORIGINAL candidate d squared := mul(290, 290)

Distance ORIGINAL candidate d squared = 84100

source roles = [290, 84100, 84100]

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.