Question 28889: Labeled Similarity at a Specified Positive Scale

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(2, 4) and B1=(3, -4), B2=(84, -4), B3=(31, -58), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=27/2. Does this scale fit every corresponding side?

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(2, 4) and B1=(3, -4), B2=(84, -4), B3=(31, -58), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=27/2. Does this scale fit every corresponding side?

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-similarity

Step-by-step derivation

source roles = [qf-similarity-specified-scale, specified_similarity]

derived = [ALL_fixed_corresponding_sides_at_given_forward_scale]

Similarity ORIGINAL scale numerator squared := mul(27, 27)

Similarity ORIGINAL scale numerator squared = 729

Similarity ORIGINAL scale denominator squared := mul(2, 2)

Similarity ORIGINAL scale denominator squared = 4

Similarity A determinant ORIGINAL point1 coordinate0 := sub(6, 0)

Similarity A determinant ORIGINAL point1 coordinate0 = 6

Similarity A determinant ORIGINAL point1 coordinate1 := sub(0, 0)

Similarity A determinant ORIGINAL point1 coordinate1 = 0

Similarity A determinant ORIGINAL point2 coordinate0 := sub(2, 0)

Similarity A determinant ORIGINAL point2 coordinate0 = 2

Similarity A determinant ORIGINAL point2 coordinate1 := sub(4, 0)

Similarity A determinant ORIGINAL point2 coordinate1 = 4

Similarity A determinant ORIGINAL first cross product := mul(6, 4)

Similarity A determinant ORIGINAL first cross product = 24

Similarity A determinant ORIGINAL second cross product := mul(0, 2)

Similarity A determinant ORIGINAL second cross product = 0

Similarity A determinant ORIGINAL cross difference := sub(24, 0)

Similarity A determinant ORIGINAL cross difference = 24

source roles = [Similarity A, [[0, 0], [6, 0], [2, 4]]]

derived = [24, Yes]

Similarity A side12 ORIGINAL coordinate0 difference := sub(6, 0)

Similarity A side12 ORIGINAL coordinate0 difference = 6

Similarity A side12 ORIGINAL coordinate1 difference := sub(0, 0)

Similarity A side12 ORIGINAL coordinate1 difference = 0

Similarity A side12 ORIGINAL coordinate0 square := mul(6, 6)

Similarity A side12 ORIGINAL coordinate0 square = 36

Similarity A side12 ORIGINAL coordinate1 square := mul(0, 0)

Similarity A side12 ORIGINAL coordinate1 square = 0

Similarity A side12 ORIGINAL squared length := sum(36, 0)

Similarity A side12 ORIGINAL squared length = 36

Similarity A side13 ORIGINAL coordinate0 difference := sub(2, 0)

Similarity A side13 ORIGINAL coordinate0 difference = 2

Similarity A side13 ORIGINAL coordinate1 difference := sub(4, 0)

Similarity A side13 ORIGINAL coordinate1 difference = 4

Similarity A side13 ORIGINAL coordinate0 square := mul(2, 2)

Similarity A side13 ORIGINAL coordinate0 square = 4

Similarity A side13 ORIGINAL coordinate1 square := mul(4, 4)

Similarity A side13 ORIGINAL coordinate1 square = 16

Similarity A side13 ORIGINAL squared length := sum(4, 16)

Similarity A side13 ORIGINAL squared length = 20

Similarity A side23 ORIGINAL coordinate0 difference := sub(2, 6)

Similarity A side23 ORIGINAL coordinate0 difference = -4

Similarity A side23 ORIGINAL coordinate1 difference := sub(4, 0)

Similarity A side23 ORIGINAL coordinate1 difference = 4

Similarity A side23 ORIGINAL coordinate0 square := mul(-4, -4)

Similarity A side23 ORIGINAL coordinate0 square = 16

Similarity A side23 ORIGINAL coordinate1 square := mul(4, 4)

Similarity A side23 ORIGINAL coordinate1 square = 16

Similarity A side23 ORIGINAL squared length := sum(16, 16)

Similarity A side23 ORIGINAL squared length = 32

Similarity B determinant ORIGINAL point1 coordinate0 := sub(84, 3)

Similarity B determinant ORIGINAL point1 coordinate0 = 81

Similarity B determinant ORIGINAL point1 coordinate1 := sub(-4, -4)

Similarity B determinant ORIGINAL point1 coordinate1 = 0

Similarity B determinant ORIGINAL point2 coordinate0 := sub(31, 3)

Similarity B determinant ORIGINAL point2 coordinate0 = 28

Similarity B determinant ORIGINAL point2 coordinate1 := sub(-58, -4)

Similarity B determinant ORIGINAL point2 coordinate1 = -54

Similarity B determinant ORIGINAL first cross product := mul(81, -54)

Similarity B determinant ORIGINAL first cross product = -4374

Similarity B determinant ORIGINAL second cross product := mul(0, 28)

Similarity B determinant ORIGINAL second cross product = 0

Similarity B determinant ORIGINAL cross difference := sub(-4374, 0)

Similarity B determinant ORIGINAL cross difference = -4374

source roles = [Similarity B, [[3, -4], [84, -4], [31, -58]]]

derived = [-4374, Yes]

Similarity B side12 ORIGINAL coordinate0 difference := sub(84, 3)

Similarity B side12 ORIGINAL coordinate0 difference = 81

Similarity B side12 ORIGINAL coordinate1 difference := sub(-4, -4)

Similarity B side12 ORIGINAL coordinate1 difference = 0

Similarity B side12 ORIGINAL coordinate0 square := mul(81, 81)

Similarity B side12 ORIGINAL coordinate0 square = 6561

Similarity B side12 ORIGINAL coordinate1 square := mul(0, 0)

Similarity B side12 ORIGINAL coordinate1 square = 0

Similarity B side12 ORIGINAL squared length := sum(6561, 0)

Similarity B side12 ORIGINAL squared length = 6561

Similarity B side13 ORIGINAL coordinate0 difference := sub(31, 3)

Similarity B side13 ORIGINAL coordinate0 difference = 28

Similarity B side13 ORIGINAL coordinate1 difference := sub(-58, -4)

Similarity B side13 ORIGINAL coordinate1 difference = -54

Similarity B side13 ORIGINAL coordinate0 square := mul(28, 28)

Similarity B side13 ORIGINAL coordinate0 square = 784

Similarity B side13 ORIGINAL coordinate1 square := mul(-54, -54)

Similarity B side13 ORIGINAL coordinate1 square = 2916

Similarity B side13 ORIGINAL squared length := sum(784, 2916)

Similarity B side13 ORIGINAL squared length = 3700

Similarity B side23 ORIGINAL coordinate0 difference := sub(31, 84)

Similarity B side23 ORIGINAL coordinate0 difference = -53

Similarity B side23 ORIGINAL coordinate1 difference := sub(-58, -4)

Similarity B side23 ORIGINAL coordinate1 difference = -54

Similarity B side23 ORIGINAL coordinate0 square := mul(-53, -53)

Similarity B side23 ORIGINAL coordinate0 square = 2809

Similarity B side23 ORIGINAL coordinate1 square := mul(-54, -54)

Similarity B side23 ORIGINAL coordinate1 square = 2916

Similarity B side23 ORIGINAL squared length := sum(2809, 2916)

Similarity B side23 ORIGINAL squared length = 5725

Similarity fixed side0 ORIGINAL Bsquare times q squared := mul(6561, 4)

Similarity fixed side0 ORIGINAL Bsquare times q squared = 26244

Similarity fixed side0 ORIGINAL Asquare times p squared := mul(36, 729)

Similarity fixed side0 ORIGINAL Asquare times p squared = 26244

source roles = [0, 36, 6561, 27, 2]

derived = [26244, 26244, Yes]

Similarity fixed side1 ORIGINAL Bsquare times q squared := mul(3700, 4)

Similarity fixed side1 ORIGINAL Bsquare times q squared = 14800

Similarity fixed side1 ORIGINAL Asquare times p squared := mul(20, 729)

Similarity fixed side1 ORIGINAL Asquare times p squared = 14580

source roles = [1, 20, 3700, 27, 2]

derived = [14800, 14580, No]

Similarity fixed side2 ORIGINAL Bsquare times q squared := mul(5725, 4)

Similarity fixed side2 ORIGINAL Bsquare times q squared = 22900

Similarity fixed side2 ORIGINAL Asquare times p squared := mul(32, 729)

Similarity fixed side2 ORIGINAL Asquare times p squared = 23328

source roles = [2, 32, 5725, 27, 2]

derived = [22900, 23328, No]

source roles = [[[0, 0], [6, 0], [2, 4]], [[3, -4], [84, -4], [31, -58]], 27, 2]

derived = [[Yes, No, No], No, No]

Reverse checks

• S059 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
• S058 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
• S057 · premise_and_result_audit — Recompute Similarity fixed side2 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Similarity fixed side2 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
• S054 · premise_and_result_audit — Recompute Similarity fixed side1 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Similarity fixed side1 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
• S051 · premise_and_result_audit — Recompute Similarity fixed side0 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Similarity fixed side0 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL squared length with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Similarity B side23 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL squared length with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Similarity B side13 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL squared length with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Similarity B side12 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
• S033 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL cross difference with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL second cross product with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL first cross product with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point2 coordinate1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point2 coordinate0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point1 coordinate1 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Similarity B determinant ORIGINAL point1 coordinate0 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL squared length with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Similarity A side23 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL squared length with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Similarity A side13 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL squared length with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Similarity A side12 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
• S010 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL cross difference with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL second cross product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL first cross product with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point2 coordinate1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point2 coordinate0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point1 coordinate1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Similarity A determinant ORIGINAL point1 coordinate0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Similarity ORIGINAL scale denominator squared with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Similarity ORIGINAL scale numerator squared with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.

Result

No

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.