Question 31355: Exact Comparison of Two Euclidean Distances

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 5). In the same frame, U=(1, -7) and V=(1, -1). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 5). In the same frame, U=(1, -7) and V=(1, -1). 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(-8, -8)

Distance PQ ORIGINAL Q minus P axis0 = 0

Distance PQ ORIGINAL Q minus P axis1 := sub(5, 1)

Distance PQ ORIGINAL Q minus P axis1 = 4

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(4, 4)

Distance PQ ORIGINAL signed delta squared axis1 = 16

Distance PQ ORIGINAL sum of two squares := sum(0, 16)

Distance PQ ORIGINAL sum of two squares = 16

source roles = [Distance PQ, [-8, 1], [-8, 5]]

derived = [[0, 4], [0, 16], 16]

Distance UV ORIGINAL Q minus P axis0 := sub(1, 1)

Distance UV ORIGINAL Q minus P axis0 = 0

Distance UV ORIGINAL Q minus P axis1 := sub(-1, -7)

Distance UV ORIGINAL Q minus P axis1 = 6

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

Distance UV ORIGINAL signed delta squared axis0 = 0

Distance UV ORIGINAL signed delta squared axis1 := mul(6, 6)

Distance UV ORIGINAL signed delta squared axis1 = 36

Distance UV ORIGINAL sum of two squares := sum(0, 36)

Distance UV ORIGINAL sum of two squares = 36

source roles = [Distance UV, [1, -7], [1, -1]]

derived = [[0, 6], [0, 36], 36]

source roles = [16, 36]

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.