Circle Boundaries and Zero-Radius Foundations
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261005-advance-20261006-r05-circle-foundations · 2026-10-05 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-12, 5) and radius r=0≥0. For P=(-12, 5), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 0 with r²=0; equality is required, giving Yes. Strictly inside points are not boundary members. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
2. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 4) and radius r=0≥0. For P=(-10, 4), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 1 with r²=0; equality is required, giving No. Strictly inside points are not boundary members. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
3. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 3) and radius r=1≥0. For P=(-8, 4), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 5 with r²=1; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
4. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 2) and radius r=2≥0. For P=(-7, 2), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 4 with r²=4; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
5. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, 1) and radius r=3≥0. For P=(-8, 3), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 4 with r²=9; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
6. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, 0) and radius r=4≥0. For P=(-2, 1), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 26 with r²=16; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
7. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. For P=(-1, -1), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 25 with r²=25; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
8. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -2) and radius r=6≥0. For P=(-5, 3), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 25 with r²=36; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
9. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -3) and radius r=7≥0. For P=(4, -2), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 65 with r²=49; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
10. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -4) and radius r=8≥0. For P=(5, -4), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 64 with r²=64; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
11. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -5) and radius r=9≥0. For P=(-2, 3), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 64 with r²=81; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
12. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. For P=(10, -5), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 122 with r²=100; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
13. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -7) and radius r=11≥0. For P=(11, -7), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 121 with r²=121; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
14. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -8) and radius r=12≥0. For P=(1, 3), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 121 with r²=144; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
15. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -9) and radius r=13≥0. For P=(16, -8), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 197 with r²=169; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
16. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -10) and radius r=14≥0. For P=(17, -10), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 196 with r²=196; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
17. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -11) and radius r=15≥0. For P=(4, 3), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 196 with r²=225; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
18. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -12) and radius r=16≥0. For P=(22, -11), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 290 with r²=256; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
19. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -13) and radius r=17≥0. For P=(23, -13), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 289 with r²=289; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
20. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -14) and radius r=18≥0. For P=(7, 3), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 289 with r²=324; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
21. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -15) and radius r=19≥0. For P=(28, -14), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 401 with r²=361; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
22. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. For P=(29, -16), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 400 with r²=400; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
23. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -17) and radius r=21≥0. For P=(10, 3), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 400 with r²=441; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
24. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -18) and radius r=22≥0. For P=(34, -17), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
No
Compare squared distance 530 with r²=484; equality is required, giving No. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
25. Membership in a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. For P=(35, -19), is P a member of the circle boundary, not merely of the filled disk?
View answer and explanation
Yes
Compare squared distance 529 with r²=529; equality is required, giving Yes. Strictly inside points are not boundary members.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate
26. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-12, 5) and radius r=0≥0. Classify P=(-12, 5) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 0 with r²=0: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
27. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 4) and radius r=0≥0. Classify P=(-10, 4) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 1 with r²=0: less means inside, equal means on boundary, and greater means outside. Here the result is outside. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
28. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 3) and radius r=1≥0. Classify P=(-8, 4) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 5 with r²=1: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
29. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 2) and radius r=2≥0. Classify P=(-7, 2) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 4 with r²=4: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
30. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, 1) and radius r=3≥0. Classify P=(-8, 3) as inside, on boundary, or outside.
View answer and explanation
inside
Compare squared distance 4 with r²=9: less means inside, equal means on boundary, and greater means outside. Here the result is inside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
31. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, 0) and radius r=4≥0. Classify P=(-2, 1) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 26 with r²=16: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
32. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. Classify P=(-1, -1) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 25 with r²=25: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
33. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -2) and radius r=6≥0. Classify P=(-5, 3) as inside, on boundary, or outside.
View answer and explanation
inside
Compare squared distance 25 with r²=36: less means inside, equal means on boundary, and greater means outside. Here the result is inside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
34. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -3) and radius r=7≥0. Classify P=(4, -2) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 65 with r²=49: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
35. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -4) and radius r=8≥0. Classify P=(5, -4) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 64 with r²=64: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
36. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -5) and radius r=9≥0. Classify P=(-2, 3) as inside, on boundary, or outside.
View answer and explanation
inside
Compare squared distance 64 with r²=81: less means inside, equal means on boundary, and greater means outside. Here the result is inside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
37. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. Classify P=(10, -5) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 122 with r²=100: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
38. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -7) and radius r=11≥0. Classify P=(11, -7) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 121 with r²=121: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
39. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -8) and radius r=12≥0. Classify P=(1, 3) as inside, on boundary, or outside.
View answer and explanation
inside
Compare squared distance 121 with r²=144: less means inside, equal means on boundary, and greater means outside. Here the result is inside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
40. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -9) and radius r=13≥0. Classify P=(16, -8) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 197 with r²=169: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
41. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -10) and radius r=14≥0. Classify P=(17, -10) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 196 with r²=196: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
42. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -11) and radius r=15≥0. Classify P=(4, 3) as inside, on boundary, or outside.
View answer and explanation
inside
Compare squared distance 196 with r²=225: less means inside, equal means on boundary, and greater means outside. Here the result is inside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
43. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -12) and radius r=16≥0. Classify P=(22, -11) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 290 with r²=256: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
44. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -13) and radius r=17≥0. Classify P=(23, -13) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 289 with r²=289: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
45. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -14) and radius r=18≥0. Classify P=(7, 3) as inside, on boundary, or outside.
View answer and explanation
inside
Compare squared distance 289 with r²=324: less means inside, equal means on boundary, and greater means outside. Here the result is inside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
46. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -15) and radius r=19≥0. Classify P=(28, -14) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 401 with r²=361: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
47. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. Classify P=(29, -16) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 400 with r²=400: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
48. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -17) and radius r=21≥0. Classify P=(10, 3) as inside, on boundary, or outside.
View answer and explanation
inside
Compare squared distance 400 with r²=441: less means inside, equal means on boundary, and greater means outside. Here the result is inside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
49. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -18) and radius r=22≥0. Classify P=(34, -17) as inside, on boundary, or outside.
View answer and explanation
outside
Compare squared distance 530 with r²=484: less means inside, equal means on boundary, and greater means outside. Here the result is outside.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
50. Three Positions Relative to a Circle Boundary
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. Classify P=(35, -19) as inside, on boundary, or outside.
View answer and explanation
on boundary
Compare squared distance 529 with r²=529: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate
51. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=0≥0. Find its diameter length.
View answer and explanation
0
The diameter length is 2r=0 in the same length unit, including zero at radius zero. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
52. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=1≥0. Find its diameter length.
View answer and explanation
2
The diameter length is 2r=2 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
53. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=2≥0. Find its diameter length.
View answer and explanation
4
The diameter length is 2r=4 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
54. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=3≥0. Find its diameter length.
View answer and explanation
6
The diameter length is 2r=6 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
55. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=4≥0. Find its diameter length.
View answer and explanation
8
The diameter length is 2r=8 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
56. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=5≥0. Find its diameter length.
View answer and explanation
10
The diameter length is 2r=10 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
57. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=6≥0. Find its diameter length.
View answer and explanation
12
The diameter length is 2r=12 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
58. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=7≥0. Find its diameter length.
View answer and explanation
14
The diameter length is 2r=14 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
59. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=8≥0. Find its diameter length.
View answer and explanation
16
The diameter length is 2r=16 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
60. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=9≥0. Find its diameter length.
View answer and explanation
18
The diameter length is 2r=18 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
61. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=10≥0. Find its diameter length.
View answer and explanation
20
The diameter length is 2r=20 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
62. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=11≥0. Find its diameter length.
View answer and explanation
22
The diameter length is 2r=22 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
63. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=12≥0. Find its diameter length.
View answer and explanation
24
The diameter length is 2r=24 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
64. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=13≥0. Find its diameter length.
View answer and explanation
26
The diameter length is 2r=26 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
65. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=14≥0. Find its diameter length.
View answer and explanation
28
The diameter length is 2r=28 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
66. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=15≥0. Find its diameter length.
View answer and explanation
30
The diameter length is 2r=30 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
67. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=16≥0. Find its diameter length.
View answer and explanation
32
The diameter length is 2r=32 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
68. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=17≥0. Find its diameter length.
View answer and explanation
34
The diameter length is 2r=34 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
69. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=18≥0. Find its diameter length.
View answer and explanation
36
The diameter length is 2r=36 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
70. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=19≥0. Find its diameter length.
View answer and explanation
38
The diameter length is 2r=38 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
71. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=20≥0. Find its diameter length.
View answer and explanation
40
The diameter length is 2r=40 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
72. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=21≥0. Find its diameter length.
View answer and explanation
42
The diameter length is 2r=42 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
73. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=22≥0. Find its diameter length.
View answer and explanation
44
The diameter length is 2r=44 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
74. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=23≥0. Find its diameter length.
View answer and explanation
46
The diameter length is 2r=46 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
75. Diameter Length for a Nonnegative Radius
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=24≥0. Find its diameter length.
View answer and explanation
48
The diameter length is 2r=48 in the same length unit, including zero at radius zero.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate
76. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, 0) and radius r=0≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(0, 0)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (0, 0), exactly on the boundary rather than inferred from a sketch. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
77. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 2) and radius r=1≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-10, 2)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-10, 2), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
78. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 1) and radius r=2≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-10, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-10, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
79. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 0) and radius r=3≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-12, 0)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, 0), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
80. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, -1) and radius r=4≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-8, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-8, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
81. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, -2) and radius r=5≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-2, -2)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-2, -2), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
82. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -3) and radius r=6≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-6, -9)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-6, -9), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
83. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -4) and radius r=7≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(2, -4)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (2, -4), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
84. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -5) and radius r=8≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-4, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-4, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
85. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -6) and radius r=9≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-12, -6)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -6), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
86. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -7) and radius r=10≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-2, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-2, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
87. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -8) and radius r=11≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(10, -8)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (10, -8), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
88. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -9) and radius r=12≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(0, -21)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (0, -21), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
89. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -10) and radius r=13≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(14, -10)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (14, -10), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
90. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -11) and radius r=14≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(2, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (2, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
91. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -12) and radius r=15≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-12, -12)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -12), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
92. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -13) and radius r=16≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(4, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (4, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
93. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -14) and radius r=17≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(22, -14)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (22, -14), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
94. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -15) and radius r=18≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(6, -33)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (6, -33), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
95. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -16) and radius r=19≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(26, -16)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (26, -16), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
96. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -17) and radius r=20≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(8, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (8, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
97. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -18) and radius r=21≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(-12, -18)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -18), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
98. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -19) and radius r=22≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(10, 3)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (10, 3), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
99. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -20) and radius r=23≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(34, -20)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (34, -20), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
100. Specified Axial Point on a Circle
Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -21) and radius r=24≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.
View answer and explanation
(12, -45)
Add the signed radius to the specified coordinate and leave the other unchanged. The point is (12, -45), exactly on the boundary rather than inferred from a sketch.
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate
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