Question 31370: Exact Comparison of Two Euclidean Distances

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-12, 6). In the same frame, U=(-14, -22) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-12, 6). In the same frame, U=(-14, -22) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.

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-comparison, distance_comparison, frame, euclidean_orthonormal_equal_units]

derived = [[PQ_versus_UV_operator]]

Distance PQ ORIGINAL Q minus P axis0 := sub(-12, 7)

Distance PQ ORIGINAL Q minus P axis0 = -19

Distance PQ ORIGINAL Q minus P axis1 := sub(6, -14)

Distance PQ ORIGINAL Q minus P axis1 = 20

Distance PQ ORIGINAL signed delta squared axis0 := mul(-19, -19)

Distance PQ ORIGINAL signed delta squared axis0 = 361

Distance PQ ORIGINAL signed delta squared axis1 := mul(20, 20)

Distance PQ ORIGINAL signed delta squared axis1 = 400

Distance PQ ORIGINAL sum of two squares := sum(361, 400)

Distance PQ ORIGINAL sum of two squares = 761

source roles = [Distance PQ, [7, -14], [-12, 6]]

derived = [[-19, 20], [361, 400], 761]

Distance UV ORIGINAL Q minus P axis0 := sub(5, -14)

Distance UV ORIGINAL Q minus P axis0 = 19

Distance UV ORIGINAL Q minus P axis1 := sub(0, -22)

Distance UV ORIGINAL Q minus P axis1 = 22

Distance UV ORIGINAL signed delta squared axis0 := mul(19, 19)

Distance UV ORIGINAL signed delta squared axis0 = 361

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

Distance UV ORIGINAL signed delta squared axis1 = 484

Distance UV ORIGINAL sum of two squares := sum(361, 484)

Distance UV ORIGINAL sum of two squares = 845

source roles = [Distance UV, [-14, -22], [5, 0]]

derived = [[19, 22], [361, 484], 845]

source roles = [761, 845]

derived = [<]

Reverse checks

• S014 · 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.
• S013 · 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.
• S012 · premise_and_result_audit — Recompute Distance UV ORIGINAL sum of two squares with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Distance UV ORIGINAL signed delta squared axis1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Distance UV ORIGINAL signed delta squared axis0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Distance UV ORIGINAL Q minus P axis1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Distance UV ORIGINAL Q minus P axis0 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

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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.