Specified Positive-Scale Triangle Similarity Foundations
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batch-20261006-advance-20261007-r06-similarity-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. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(6, -4), B3=(4, -6), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=1/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[9, 5, 8]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
2. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(9, -4), B3=(5, -8), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=3/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[36, 20, 32]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
3. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(12, -4), B3=(6, -10), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=3/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[81, 45, 72]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
4. Labeled Similarity at a Specified Positive Scale
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=(24, -4), B3=(10, -18), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=7/2. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[36, 20, 32] and B=[441, 245, 392]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
5. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(18, -4), B3=(8, -14), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=5/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[225, 125, 200]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
6. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(21, -4), B3=(9, -16), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=7/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[324, 180, 288]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
7. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(24, -4), B3=(11, -18), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=7/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[441, 260, 365]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
8. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(27, -4), B3=(11, -20), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=9/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[576, 320, 512]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
9. Labeled Similarity at a Specified Positive Scale
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=(54, -4), B3=(20, -38), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/2. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[36, 20, 32] and B=[2601, 1445, 2312]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
10. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(33, -4), B3=(13, -24), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=11/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[900, 500, 800]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
11. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(36, -4), B3=(14, -26), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=11/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[1089, 605, 968]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
12. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(39, -4), B3=(15, -28), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=13/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[1296, 720, 1152]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
13. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(42, -4), B3=(16, -30), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=13/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[1521, 845, 1352]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
14. Labeled Similarity at a Specified Positive Scale
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?
View answer and explanation
No
The squared sides are A=[36, 20, 32] and B=[6561, 3700, 5725]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
15. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(48, -4), B3=(18, -34), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=15/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[2025, 1125, 1800]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
16. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(51, -4), B3=(19, -36), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[2304, 1280, 2048]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
17. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(54, -4), B3=(20, -38), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=17/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[2601, 1445, 2312]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
18. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(57, -4), B3=(21, -40), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=19/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[2916, 1620, 2592]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
19. Labeled Similarity at a Specified Positive Scale
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?
View answer and explanation
Yes
The squared sides are A=[36, 20, 32] and B=[12321, 6845, 10952]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
20. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(63, -4), B3=(23, -44), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=21/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[3600, 2000, 3200]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
21. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(66, -4), B3=(25, -46), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=21/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[3969, 2248, 3445]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
22. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(69, -4), B3=(25, -48), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=23/1. Does this scale fit every corresponding side?
View answer and explanation
No
The squared sides are A=[9, 5, 8] and B=[4356, 2420, 3872]. Check B²q²=A²p² on each corresponding side, obtaining No. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
23. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(72, -4), B3=(26, -50), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=23/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[4761, 2645, 4232]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
24. Labeled Similarity at a Specified Positive Scale
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=(144, -4), B3=(50, -98), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=47/2. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[36, 20, 32] and B=[19881, 11045, 17672]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
25. Labeled Similarity at a Specified Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(78, -4), B3=(28, -54), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=25/1. Does this scale fit every corresponding side?
View answer and explanation
Yes
The squared sides are A=[9, 5, 8] and B=[5625, 3125, 5000]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-specified-scale · candidate
26. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-12/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
27. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-11/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
28. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-10/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
29. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-9/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
30. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-8/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
31. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-7/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
32. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-6/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
33. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-5/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
34. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-4/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
35. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-3/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
36. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-2/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
37. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=-1/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
38. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=0/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
No
The denominator is valid, and the numerator must also be positive, giving No. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
39. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=1/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
40. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=2/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
41. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=3/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
42. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=4/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
43. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=5/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
44. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=6/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
45. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=7/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
46. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=8/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
47. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=9/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
48. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=10/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
49. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=11/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
50. Positive Length-Scale Eligibility
Use one equal-unit orthonormal Euclidean Cartesian frame. With positive denominator q=1, is k=12/1 an eligible positive length scale for nondegenerate similarity?
View answer and explanation
Yes
The denominator is valid, and the numerator must also be positive, giving Yes. Zero or negative candidates fail this positive-length-scale convention; do not reinterpret them as signed coordinate coefficients.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-positive-ratio · candidate
51. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 2>0. At corresponding positive length scale k=1/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
2/1
Multiply the given positive squared length by k² to obtain 2/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
52. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 5>0. At corresponding positive length scale k=2/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
5/1
Multiply the given positive squared length by k² to obtain 5/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
53. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 10>0. At corresponding positive length scale k=3/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
10/1
Multiply the given positive squared length by k² to obtain 10/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
54. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 17>0. At corresponding positive length scale k=4/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
272/1
Multiply the given positive squared length by k² to obtain 272/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
55. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 26>0. At corresponding positive length scale k=5/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
325/2
Multiply the given positive squared length by k² to obtain 325/2. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
56. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 37>0. At corresponding positive length scale k=1/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
37/9
Multiply the given positive squared length by k² to obtain 37/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
57. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 50>0. At corresponding positive length scale k=2/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
200/1
Multiply the given positive squared length by k² to obtain 200/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
58. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 65>0. At corresponding positive length scale k=3/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
585/4
Multiply the given positive squared length by k² to obtain 585/4. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
59. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 82>0. At corresponding positive length scale k=4/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
1312/9
Multiply the given positive squared length by k² to obtain 1312/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
60. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 101>0. At corresponding positive length scale k=5/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
2525/1
Multiply the given positive squared length by k² to obtain 2525/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
61. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 122>0. At corresponding positive length scale k=1/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
61/2
Multiply the given positive squared length by k² to obtain 61/2. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
62. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 145>0. At corresponding positive length scale k=2/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
580/9
Multiply the given positive squared length by k² to obtain 580/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
63. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 170>0. At corresponding positive length scale k=3/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
1530/1
Multiply the given positive squared length by k² to obtain 1530/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
64. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 197>0. At corresponding positive length scale k=4/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
788/1
Multiply the given positive squared length by k² to obtain 788/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
65. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 226>0. At corresponding positive length scale k=5/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
5650/9
Multiply the given positive squared length by k² to obtain 5650/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
66. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 257>0. At corresponding positive length scale k=1/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
257/1
Multiply the given positive squared length by k² to obtain 257/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
67. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 290>0. At corresponding positive length scale k=2/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
290/1
Multiply the given positive squared length by k² to obtain 290/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
68. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 325>0. At corresponding positive length scale k=3/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
325/1
Multiply the given positive squared length by k² to obtain 325/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
69. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 362>0. At corresponding positive length scale k=4/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
5792/1
Multiply the given positive squared length by k² to obtain 5792/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
70. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 401>0. At corresponding positive length scale k=5/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
10025/4
Multiply the given positive squared length by k² to obtain 10025/4. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
71. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 442>0. At corresponding positive length scale k=1/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
442/9
Multiply the given positive squared length by k² to obtain 442/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
72. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 485>0. At corresponding positive length scale k=2/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
1940/1
Multiply the given positive squared length by k² to obtain 1940/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
73. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 530>0. At corresponding positive length scale k=3/2, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
2385/2
Multiply the given positive squared length by k² to obtain 2385/2. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
74. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 577>0. At corresponding positive length scale k=4/3, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
9232/9
Multiply the given positive squared length by k² to obtain 9232/9. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
75. Squared Side Length under a Positive Scale
Use one equal-unit orthonormal Euclidean Cartesian frame. A side of a nondegenerate triangle has given squared length 626>0. At corresponding positive length scale k=5/1, find its scaled squared length. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
15650/1
Multiply the given positive squared length by k² to obtain 15650/1. Length scales by k, but squared length scales by k²; this is exact conversion of the supplied quantity.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-squared-side · candidate
76. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 1/2>0 squared units. At corresponding positive length scale k=1/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
1/2
Multiply the given positive area by k² to obtain 1/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
77. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 3/2>0 squared units. At corresponding positive length scale k=2/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
3/2
Multiply the given positive area by k² to obtain 3/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
78. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 5/2>0 squared units. At corresponding positive length scale k=3/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
5/2
Multiply the given positive area by k² to obtain 5/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
79. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 7/2>0 squared units. At corresponding positive length scale k=4/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
56/1
Multiply the given positive area by k² to obtain 56/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
80. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 9/2>0 squared units. At corresponding positive length scale k=1/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
9/8
Multiply the given positive area by k² to obtain 9/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
81. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 11/2>0 squared units. At corresponding positive length scale k=2/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
22/9
Multiply the given positive area by k² to obtain 22/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
82. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 13/2>0 squared units. At corresponding positive length scale k=3/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
117/2
Multiply the given positive area by k² to obtain 117/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
83. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 15/2>0 squared units. At corresponding positive length scale k=4/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
30/1
Multiply the given positive area by k² to obtain 30/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
84. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 17/2>0 squared units. At corresponding positive length scale k=1/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
17/18
Multiply the given positive area by k² to obtain 17/18 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
85. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 19/2>0 squared units. At corresponding positive length scale k=2/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
38/1
Multiply the given positive area by k² to obtain 38/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
86. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 21/2>0 squared units. At corresponding positive length scale k=3/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
189/8
Multiply the given positive area by k² to obtain 189/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
87. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 23/2>0 squared units. At corresponding positive length scale k=4/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
184/9
Multiply the given positive area by k² to obtain 184/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
88. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 25/2>0 squared units. At corresponding positive length scale k=1/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
25/2
Multiply the given positive area by k² to obtain 25/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
89. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 27/2>0 squared units. At corresponding positive length scale k=2/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
27/2
Multiply the given positive area by k² to obtain 27/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
90. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 29/2>0 squared units. At corresponding positive length scale k=3/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
29/2
Multiply the given positive area by k² to obtain 29/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
91. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 31/2>0 squared units. At corresponding positive length scale k=4/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
248/1
Multiply the given positive area by k² to obtain 248/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
92. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 33/2>0 squared units. At corresponding positive length scale k=1/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
33/8
Multiply the given positive area by k² to obtain 33/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
93. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 35/2>0 squared units. At corresponding positive length scale k=2/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
70/9
Multiply the given positive area by k² to obtain 70/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
94. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 37/2>0 squared units. At corresponding positive length scale k=3/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
333/2
Multiply the given positive area by k² to obtain 333/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
95. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 39/2>0 squared units. At corresponding positive length scale k=4/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
View answer and explanation
78/1
Multiply the given positive area by k² to obtain 78/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
96. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 41/2>0 squared units. At corresponding positive length scale k=1/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
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41/18
Multiply the given positive area by k² to obtain 41/18 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
97. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 43/2>0 squared units. At corresponding positive length scale k=2/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
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86/1
Multiply the given positive area by k² to obtain 86/1 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
98. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 45/2>0 squared units. At corresponding positive length scale k=3/2, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
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405/8
Multiply the given positive area by k² to obtain 405/8 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
99. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 47/2>0 squared units. At corresponding positive length scale k=4/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
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376/9
Multiply the given positive area by k² to obtain 376/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
100. Conditional Scaling of a Given Positive Area
Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 49/2>0 squared units. At corresponding positive length scale k=1/1, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.
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49/2
Multiply the given positive area by k² to obtain 49/2 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory.
Reviewed step-by-step derivation
Labeled triangle similarity with a specified positive scale · qf-similarity-scaled-area · candidate
Canonical question JSONL · English JSONL · Validation report