Question 28894: 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=(114, -4), B3=(40, -78), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=37/2. 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=(114, -4), B3=(40, -78), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=37/2. 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(37, 37)

Similarity ORIGINAL scale numerator squared = 1369

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(114, 3)

Similarity B determinant ORIGINAL point1 coordinate0 = 111

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

Similarity B determinant ORIGINAL point1 coordinate1 = 0

Similarity B determinant ORIGINAL point2 coordinate0 := sub(40, 3)

Similarity B determinant ORIGINAL point2 coordinate0 = 37

Similarity B determinant ORIGINAL point2 coordinate1 := sub(-78, -4)

Similarity B determinant ORIGINAL point2 coordinate1 = -74

Similarity B determinant ORIGINAL first cross product := mul(111, -74)

Similarity B determinant ORIGINAL first cross product = -8214

Similarity B determinant ORIGINAL second cross product := mul(0, 37)

Similarity B determinant ORIGINAL second cross product = 0

Similarity B determinant ORIGINAL cross difference := sub(-8214, 0)

Similarity B determinant ORIGINAL cross difference = -8214

source roles = [Similarity B, [[3, -4], [114, -4], [40, -78]]]

derived = [-8214, Yes]

Similarity B side12 ORIGINAL coordinate0 difference := sub(114, 3)

Similarity B side12 ORIGINAL coordinate0 difference = 111

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

Similarity B side12 ORIGINAL coordinate1 difference = 0

Similarity B side12 ORIGINAL coordinate0 square := mul(111, 111)

Similarity B side12 ORIGINAL coordinate0 square = 12321

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

Similarity B side12 ORIGINAL coordinate1 square = 0

Similarity B side12 ORIGINAL squared length := sum(12321, 0)

Similarity B side12 ORIGINAL squared length = 12321

Similarity B side13 ORIGINAL coordinate0 difference := sub(40, 3)

Similarity B side13 ORIGINAL coordinate0 difference = 37

Similarity B side13 ORIGINAL coordinate1 difference := sub(-78, -4)

Similarity B side13 ORIGINAL coordinate1 difference = -74

Similarity B side13 ORIGINAL coordinate0 square := mul(37, 37)

Similarity B side13 ORIGINAL coordinate0 square = 1369

Similarity B side13 ORIGINAL coordinate1 square := mul(-74, -74)

Similarity B side13 ORIGINAL coordinate1 square = 5476

Similarity B side13 ORIGINAL squared length := sum(1369, 5476)

Similarity B side13 ORIGINAL squared length = 6845

Similarity B side23 ORIGINAL coordinate0 difference := sub(40, 114)

Similarity B side23 ORIGINAL coordinate0 difference = -74

Similarity B side23 ORIGINAL coordinate1 difference := sub(-78, -4)

Similarity B side23 ORIGINAL coordinate1 difference = -74

Similarity B side23 ORIGINAL coordinate0 square := mul(-74, -74)

Similarity B side23 ORIGINAL coordinate0 square = 5476

Similarity B side23 ORIGINAL coordinate1 square := mul(-74, -74)

Similarity B side23 ORIGINAL coordinate1 square = 5476

Similarity B side23 ORIGINAL squared length := sum(5476, 5476)

Similarity B side23 ORIGINAL squared length = 10952

Similarity fixed side0 ORIGINAL Bsquare times q squared := mul(12321, 4)

Similarity fixed side0 ORIGINAL Bsquare times q squared = 49284

Similarity fixed side0 ORIGINAL Asquare times p squared := mul(36, 1369)

Similarity fixed side0 ORIGINAL Asquare times p squared = 49284

source roles = [0, 36, 12321, 37, 2]

derived = [49284, 49284, Yes]

Similarity fixed side1 ORIGINAL Bsquare times q squared := mul(6845, 4)

Similarity fixed side1 ORIGINAL Bsquare times q squared = 27380

Similarity fixed side1 ORIGINAL Asquare times p squared := mul(20, 1369)

Similarity fixed side1 ORIGINAL Asquare times p squared = 27380

source roles = [1, 20, 6845, 37, 2]

derived = [27380, 27380, Yes]

Similarity fixed side2 ORIGINAL Bsquare times q squared := mul(10952, 4)

Similarity fixed side2 ORIGINAL Bsquare times q squared = 43808

Similarity fixed side2 ORIGINAL Asquare times p squared := mul(32, 1369)

Similarity fixed side2 ORIGINAL Asquare times p squared = 43808

source roles = [2, 32, 10952, 37, 2]

derived = [43808, 43808, Yes]

source roles = [[[0, 0], [6, 0], [2, 4]], [[3, -4], [114, -4], [40, -78]], 37, 2]

derived = [[Yes, Yes, Yes], Yes, Yes]

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

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.