Labeled Nondegenerate Triangle Congruence Foundations
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batch-20261006-advance-20261007-r05-congruence-foundations · 2026-10-06 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(1, 0), A3=(1, 2) and B1=(3, 5), B2=(4, 5), B3=(4, 3), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
Yes
In the fixed order 12,13,23, squared sides are A=[1, 5, 4] and B=[1, 5, 4]. Compare all three to obtain Yes. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
2. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(2, 0), A3=(1, 3) and B1=(0, 0), B2=(3, 0), B3=(1, 3), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[4, 10, 10] and B=[9, 10, 13]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
3. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 4) and B1=(0, 0), B2=(3, 0), B3=(1, 5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[9, 17, 20] and B=[9, 26, 29]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
4. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(4, 0), A3=(1, 5) and B1=(0, 0), B2=(4, 0), B3=(-1, 5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[16, 26, 34] and B=[16, 26, 50]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
5. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(5, 0), A3=(1, 6) and B1=(3, 5), B2=(8, 5), B3=(4, -1), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
Yes
In the fixed order 12,13,23, squared sides are A=[25, 37, 52] and B=[25, 37, 52]. Compare all three to obtain Yes. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
6. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(1, 7) and B1=(0, 0), B2=(7, 0), B3=(1, 7), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[36, 50, 74] and B=[49, 50, 85]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
7. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(7, 0), A3=(1, 8) and B1=(0, 0), B2=(7, 0), B3=(1, 9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[49, 65, 100] and B=[49, 82, 117]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
8. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(8, 0), A3=(1, 9) and B1=(0, 0), B2=(8, 0), B3=(-1, 9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[64, 82, 130] and B=[64, 82, 162]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
9. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(9, 0), A3=(1, 10) and B1=(3, 5), B2=(12, 5), B3=(4, -5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
Yes
In the fixed order 12,13,23, squared sides are A=[81, 101, 164] and B=[81, 101, 164]. Compare all three to obtain Yes. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
10. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(10, 0), A3=(1, 11) and B1=(0, 0), B2=(11, 0), B3=(1, 11), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[100, 122, 202] and B=[121, 122, 221]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
11. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(11, 0), A3=(1, 12) and B1=(0, 0), B2=(11, 0), B3=(1, 13), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[121, 145, 244] and B=[121, 170, 269]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
12. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(12, 0), A3=(1, 13) and B1=(0, 0), B2=(12, 0), B3=(-1, 13), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[144, 170, 290] and B=[144, 170, 338]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
13. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(13, 0), A3=(1, 14) and B1=(3, 5), B2=(16, 5), B3=(4, -9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
Yes
In the fixed order 12,13,23, squared sides are A=[169, 197, 340] and B=[169, 197, 340]. Compare all three to obtain Yes. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
14. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(14, 0), A3=(1, 15) and B1=(0, 0), B2=(15, 0), B3=(1, 15), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[196, 226, 394] and B=[225, 226, 421]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
15. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(15, 0), A3=(1, 16) and B1=(0, 0), B2=(15, 0), B3=(1, 17), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[225, 257, 452] and B=[225, 290, 485]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
16. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(16, 0), A3=(1, 17) and B1=(0, 0), B2=(16, 0), B3=(-1, 17), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[256, 290, 514] and B=[256, 290, 578]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
17. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(17, 0), A3=(1, 18) and B1=(3, 5), B2=(20, 5), B3=(4, -13), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
Yes
In the fixed order 12,13,23, squared sides are A=[289, 325, 580] and B=[289, 325, 580]. Compare all three to obtain Yes. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
18. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(18, 0), A3=(1, 19) and B1=(0, 0), B2=(19, 0), B3=(1, 19), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[324, 362, 650] and B=[361, 362, 685]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
19. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(19, 0), A3=(1, 20) and B1=(0, 0), B2=(19, 0), B3=(1, 21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[361, 401, 724] and B=[361, 442, 765]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
20. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(20, 0), A3=(1, 21) and B1=(0, 0), B2=(20, 0), B3=(-1, 21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[400, 442, 802] and B=[400, 442, 882]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
21. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(21, 0), A3=(1, 22) and B1=(3, 5), B2=(24, 5), B3=(4, -17), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
Yes
In the fixed order 12,13,23, squared sides are A=[441, 485, 884] and B=[441, 485, 884]. Compare all three to obtain Yes. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
22. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(22, 0), A3=(1, 23) and B1=(0, 0), B2=(23, 0), B3=(1, 23), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[484, 530, 970] and B=[529, 530, 1013]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
23. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(23, 0), A3=(1, 24) and B1=(0, 0), B2=(23, 0), B3=(1, 25), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[529, 577, 1060] and B=[529, 626, 1109]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
24. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(24, 0), A3=(1, 25) and B1=(0, 0), B2=(24, 0), B3=(-1, 25), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
No
In the fixed order 12,13,23, squared sides are A=[576, 626, 1154] and B=[576, 626, 1250]. Compare all three to obtain No. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
25. SSS Congruence with Fixed Vertex Labels
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(25, 0), A3=(1, 26) and B1=(3, 5), B2=(28, 5), B3=(4, -21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?
View answer and explanation
Yes
In the fixed order 12,13,23, squared sides are A=[625, 677, 1252] and B=[625, 677, 1252]. Compare all three to obtain Yes. Reflection is allowed; failure of this correspondence does not reject every other vertex permutation.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-labeled-sss · candidate
26. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(0, 1), A3=(-1, 0) and B1=(3, 5), B2=(3, 4), B3=(2, 5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
1
Read the specified side after the complete labeled-congruence check, obtaining 1. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
27. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(1, 2), A3=(-2, 1) and B1=(3, 5), B2=(4, 3), B3=(1, 4), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
5
Read the specified side after the complete labeled-congruence check, obtaining 5. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
28. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(2, 3), A3=(-3, 2) and B1=(3, 5), B2=(5, 2), B3=(0, 3), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
13
Read the specified side after the complete labeled-congruence check, obtaining 13. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
29. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 4), A3=(-4, 3) and B1=(3, 5), B2=(6, 1), B3=(-1, 2), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
25
Read the specified side after the complete labeled-congruence check, obtaining 25. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
30. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(4, 5), A3=(-5, 4) and B1=(3, 5), B2=(7, 0), B3=(-2, 1), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
41
Read the specified side after the complete labeled-congruence check, obtaining 41. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
31. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(5, 6), A3=(-6, 5) and B1=(3, 5), B2=(8, -1), B3=(-3, 0), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
61
Read the specified side after the complete labeled-congruence check, obtaining 61. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
32. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 7), A3=(-7, 6) and B1=(3, 5), B2=(9, -2), B3=(-4, -1), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
85
Read the specified side after the complete labeled-congruence check, obtaining 85. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
33. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(7, 8), A3=(-8, 7) and B1=(3, 5), B2=(10, -3), B3=(-5, -2), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
113
Read the specified side after the complete labeled-congruence check, obtaining 113. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
34. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(8, 9), A3=(-9, 8) and B1=(3, 5), B2=(11, -4), B3=(-6, -3), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
145
Read the specified side after the complete labeled-congruence check, obtaining 145. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
35. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(9, 10), A3=(-10, 9) and B1=(3, 5), B2=(12, -5), B3=(-7, -4), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
181
Read the specified side after the complete labeled-congruence check, obtaining 181. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
36. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(10, 11), A3=(-11, 10) and B1=(3, 5), B2=(13, -6), B3=(-8, -5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
221
Read the specified side after the complete labeled-congruence check, obtaining 221. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
37. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(11, 12), A3=(-12, 11) and B1=(3, 5), B2=(14, -7), B3=(-9, -6), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
265
Read the specified side after the complete labeled-congruence check, obtaining 265. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
38. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(12, 13), A3=(-13, 12) and B1=(3, 5), B2=(15, -8), B3=(-10, -7), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
313
Read the specified side after the complete labeled-congruence check, obtaining 313. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
39. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(13, 14), A3=(-14, 13) and B1=(3, 5), B2=(16, -9), B3=(-11, -8), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
365
Read the specified side after the complete labeled-congruence check, obtaining 365. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
40. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(14, 15), A3=(-15, 14) and B1=(3, 5), B2=(17, -10), B3=(-12, -9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
421
Read the specified side after the complete labeled-congruence check, obtaining 421. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
41. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(15, 16), A3=(-16, 15) and B1=(3, 5), B2=(18, -11), B3=(-13, -10), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
481
Read the specified side after the complete labeled-congruence check, obtaining 481. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
42. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(16, 17), A3=(-17, 16) and B1=(3, 5), B2=(19, -12), B3=(-14, -11), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
545
Read the specified side after the complete labeled-congruence check, obtaining 545. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
43. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(17, 18), A3=(-18, 17) and B1=(3, 5), B2=(20, -13), B3=(-15, -12), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
613
Read the specified side after the complete labeled-congruence check, obtaining 613. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
44. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(18, 19), A3=(-19, 18) and B1=(3, 5), B2=(21, -14), B3=(-16, -13), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
685
Read the specified side after the complete labeled-congruence check, obtaining 685. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
45. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(19, 20), A3=(-20, 19) and B1=(3, 5), B2=(22, -15), B3=(-17, -14), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
761
Read the specified side after the complete labeled-congruence check, obtaining 761. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
46. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(20, 21), A3=(-21, 20) and B1=(3, 5), B2=(23, -16), B3=(-18, -15), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
841
Read the specified side after the complete labeled-congruence check, obtaining 841. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
47. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(21, 22), A3=(-22, 21) and B1=(3, 5), B2=(24, -17), B3=(-19, -16), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
925
Read the specified side after the complete labeled-congruence check, obtaining 925. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
48. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(22, 23), A3=(-23, 22) and B1=(3, 5), B2=(25, -18), B3=(-20, -17), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
1013
Read the specified side after the complete labeled-congruence check, obtaining 1013. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
49. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(23, 24), A3=(-24, 23) and B1=(3, 5), B2=(26, -19), B3=(-21, -18), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
1105
Read the specified side after the complete labeled-congruence check, obtaining 1105. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
50. Corresponding Squared Side in Congruent Triangles
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(24, 25), A3=(-25, 24) and B1=(3, 5), B2=(27, -20), B3=(-22, -19), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Full three-side congruence has been confirmed. Find the common squared length of A1A2 and B1B2.
View answer and explanation
1201
Read the specified side after the complete labeled-congruence check, obtaining 1201. Agreement of just one side alone would not establish congruence.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-corresponding-squared-side · candidate
51. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(1, 0), A3=(1, 2) and B1=(3, 5), B2=(4, 5), B3=(4, 3), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
none
The ordered squared sides are A=[1, 5, 4] and B=[1, 5, 4]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
52. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(2, 0), A3=(1, 3) and B1=(0, 0), B2=(3, 0), B3=(1, 3), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
12
The ordered squared sides are A=[4, 10, 10] and B=[9, 10, 13]. The first mismatch is 12; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
53. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 4) and B1=(0, 0), B2=(3, 0), B3=(1, 5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
13
The ordered squared sides are A=[9, 17, 20] and B=[9, 26, 29]. The first mismatch is 13; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
54. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(4, 0), A3=(1, 5) and B1=(0, 0), B2=(4, 0), B3=(-1, 5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
23
The ordered squared sides are A=[16, 26, 34] and B=[16, 26, 50]. The first mismatch is 23; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
55. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(5, 0), A3=(1, 6) and B1=(3, 5), B2=(8, 5), B3=(4, -1), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
none
The ordered squared sides are A=[25, 37, 52] and B=[25, 37, 52]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
56. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(6, 0), A3=(1, 7) and B1=(0, 0), B2=(7, 0), B3=(1, 7), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
12
The ordered squared sides are A=[36, 50, 74] and B=[49, 50, 85]. The first mismatch is 12; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
57. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(7, 0), A3=(1, 8) and B1=(0, 0), B2=(7, 0), B3=(1, 9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
13
The ordered squared sides are A=[49, 65, 100] and B=[49, 82, 117]. The first mismatch is 13; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
58. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(8, 0), A3=(1, 9) and B1=(0, 0), B2=(8, 0), B3=(-1, 9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
23
The ordered squared sides are A=[64, 82, 130] and B=[64, 82, 162]. The first mismatch is 23; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
59. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(9, 0), A3=(1, 10) and B1=(3, 5), B2=(12, 5), B3=(4, -5), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
none
The ordered squared sides are A=[81, 101, 164] and B=[81, 101, 164]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
60. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(10, 0), A3=(1, 11) and B1=(0, 0), B2=(11, 0), B3=(1, 11), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
12
The ordered squared sides are A=[100, 122, 202] and B=[121, 122, 221]. The first mismatch is 12; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
61. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(11, 0), A3=(1, 12) and B1=(0, 0), B2=(11, 0), B3=(1, 13), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
13
The ordered squared sides are A=[121, 145, 244] and B=[121, 170, 269]. The first mismatch is 13; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
62. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(12, 0), A3=(1, 13) and B1=(0, 0), B2=(12, 0), B3=(-1, 13), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
23
The ordered squared sides are A=[144, 170, 290] and B=[144, 170, 338]. The first mismatch is 23; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
63. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(13, 0), A3=(1, 14) and B1=(3, 5), B2=(16, 5), B3=(4, -9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
none
The ordered squared sides are A=[169, 197, 340] and B=[169, 197, 340]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
64. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(14, 0), A3=(1, 15) and B1=(0, 0), B2=(15, 0), B3=(1, 15), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
12
The ordered squared sides are A=[196, 226, 394] and B=[225, 226, 421]. The first mismatch is 12; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
65. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(15, 0), A3=(1, 16) and B1=(0, 0), B2=(15, 0), B3=(1, 17), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
13
The ordered squared sides are A=[225, 257, 452] and B=[225, 290, 485]. The first mismatch is 13; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
66. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(16, 0), A3=(1, 17) and B1=(0, 0), B2=(16, 0), B3=(-1, 17), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
23
The ordered squared sides are A=[256, 290, 514] and B=[256, 290, 578]. The first mismatch is 23; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
67. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(17, 0), A3=(1, 18) and B1=(3, 5), B2=(20, 5), B3=(4, -13), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
none
The ordered squared sides are A=[289, 325, 580] and B=[289, 325, 580]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
68. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(18, 0), A3=(1, 19) and B1=(0, 0), B2=(19, 0), B3=(1, 19), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
12
The ordered squared sides are A=[324, 362, 650] and B=[361, 362, 685]. The first mismatch is 12; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
69. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(19, 0), A3=(1, 20) and B1=(0, 0), B2=(19, 0), B3=(1, 21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
13
The ordered squared sides are A=[361, 401, 724] and B=[361, 442, 765]. The first mismatch is 13; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
70. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(20, 0), A3=(1, 21) and B1=(0, 0), B2=(20, 0), B3=(-1, 21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
23
The ordered squared sides are A=[400, 442, 802] and B=[400, 442, 882]. The first mismatch is 23; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
71. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(21, 0), A3=(1, 22) and B1=(3, 5), B2=(24, 5), B3=(4, -17), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
none
The ordered squared sides are A=[441, 485, 884] and B=[441, 485, 884]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
72. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(22, 0), A3=(1, 23) and B1=(0, 0), B2=(23, 0), B3=(1, 23), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
12
The ordered squared sides are A=[484, 530, 970] and B=[529, 530, 1013]. The first mismatch is 12; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
73. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(23, 0), A3=(1, 24) and B1=(0, 0), B2=(23, 0), B3=(1, 25), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
13
The ordered squared sides are A=[529, 577, 1060] and B=[529, 626, 1109]. The first mismatch is 13; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
74. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(24, 0), A3=(1, 25) and B1=(0, 0), B2=(24, 0), B3=(-1, 25), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
23
The ordered squared sides are A=[576, 626, 1154] and B=[576, 626, 1250]. The first mismatch is 23; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
75. First Mismatch in Fixed Corresponding Sides
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(25, 0), A3=(1, 26) and B1=(3, 5), B2=(28, 5), B3=(4, -21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.
View answer and explanation
none
The ordered squared sides are A=[625, 677, 1252] and B=[625, 677, 1252]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-first-mismatch · candidate
76. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(0, 0), A3=(0, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
77. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(0, 0), A3=(1, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
78. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(1, 0), A3=(2, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
79. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(3, 0), A3=(1, 4). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
80. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(4, 0), A3=(5, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
81. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(5, 0), A3=(5, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
82. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(6, 0), A3=(1, 7). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
83. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(7, 0), A3=(8, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
84. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(8, 0), A3=(8, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
85. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(9, 0), A3=(1, 10). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
86. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(10, 0), A3=(11, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
87. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(11, 0), A3=(11, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
88. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(12, 0), A3=(1, 13). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
89. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(13, 0), A3=(14, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
90. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(14, 0), A3=(14, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
91. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(15, 0), A3=(1, 16). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
92. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(16, 0), A3=(17, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
93. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(17, 0), A3=(17, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
94. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(18, 0), A3=(1, 19). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
95. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(19, 0), A3=(20, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
96. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(20, 0), A3=(20, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
97. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(21, 0), A3=(1, 22). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
98. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(22, 0), A3=(23, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
99. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(23, 0), A3=(23, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
No
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
100. Triangle Eligibility of an Ordered Point Triple
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(24, 0), A3=(1, 25). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.
View answer and explanation
Yes
Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving Yes. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple.
Reviewed step-by-step derivation
Congruence of nondegenerate triangles with explicit correspondence · qf-congruence-triangle-validity · candidate
Canonical question JSONL · English JSONL · Validation report