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Question 31360: Exact Comparison of Two Euclidean Distances

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0061-31360 · q-20261005-advance-20261006-r04-distance-formula-0085

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-12, 6). In the same frame, U=(-4, -12) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.

  • G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-12, 6). In the same frame, U=(-4, -12) and V=(5, -2). 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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-distance-comparison, distance_comparison, frame, euclidean_orthonormal_equal_units]

After

derived = [[PQ_versus_UV_operator]]

Conditions: none

Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.

Step identifiers and dependencies

Rule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001

S002 · calculation

Before

Distance PQ ORIGINAL Q minus P axis0 := sub(-12, -3)

After

Distance PQ ORIGINAL Q minus P axis0 = -9

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Distance PQ ORIGINAL Q minus P axis1 := sub(6, -4)

After

Distance PQ ORIGINAL Q minus P axis1 = 10

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Distance PQ ORIGINAL signed delta squared axis0 := mul(-9, -9)

After

Distance PQ ORIGINAL signed delta squared axis0 = 81

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Distance PQ ORIGINAL signed delta squared axis1 := mul(10, 10)

After

Distance PQ ORIGINAL signed delta squared axis1 = 100

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Distance PQ ORIGINAL sum of two squares := sum(81, 100)

After

Distance PQ ORIGINAL sum of two squares = 181

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Distance PQ, [-3, -4], [-12, 6]]

After

derived = [[-9, 10], [81, 100], 181]

Conditions: none

In the DECLARED orthonormal equal-unit frame,distance squared is(Δx)²+(Δy)²≥0,including coincident points.Keep signed differences;do not mistake a signed component for distance.

Step identifiers and dependencies

Rule: Distance_ORTHONORMAL_equal_units_two_squares_NONNEGATIVE · Uses: G001, S006

S008 · calculation

Before

Distance UV ORIGINAL Q minus P axis0 := sub(5, -4)

After

Distance UV ORIGINAL Q minus P axis0 = 9

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Distance UV ORIGINAL Q minus P axis1 := sub(-2, -12)

After

Distance UV ORIGINAL Q minus P axis1 = 10

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Distance UV ORIGINAL signed delta squared axis0 := mul(9, 9)

After

Distance UV ORIGINAL signed delta squared axis0 = 81

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Distance UV ORIGINAL signed delta squared axis1 := mul(10, 10)

After

Distance UV ORIGINAL signed delta squared axis1 = 100

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Distance UV ORIGINAL sum of two squares := sum(81, 100)

After

Distance UV ORIGINAL sum of two squares = 181

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

source roles = [Distance UV, [-4, -12], [5, -2]]

After

derived = [[9, 10], [81, 100], 181]

Conditions: none

In the DECLARED orthonormal equal-unit frame,distance squared is(Δx)²+(Δy)²≥0,including coincident points.Keep signed differences;do not mistake a signed component for distance.

Step identifiers and dependencies

Rule: Distance_ORTHONORMAL_equal_units_two_squares_NONNEGATIVE · Uses: G001, S012

S014 · implies

Before

source roles = [181, 181]

After

derived = [=]

Conditions: none

Both distances are nonnegative,and the principal square root is strictly increasing there,so compare SQUARED norms in PQ-versus-UV order.Neither norm need be a perfect square;do not compare signed components or coordinate order.

Step identifiers and dependencies

Rule: Distance_COMPARE_PQ_versus_UV_by_nondecreasing_principal_roots · Uses: G001, S013

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

=

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 4db9611f2ad7abf06e91949dd352dd8e54e645e9406baaed0ccbc98c02026cf6
  • candidate record SHA-256: 6c5a0a3d5063253e3f344130917e4c149d4d633df31ed9b91fc720d1576b0a80
  • semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69
  • question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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