Directed Components, Principal Lengths, and Midpoints
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261008-advance-20261012-r06-euclidean-length-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[0, 0]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(0, 0, 0)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (0, 0, 0).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
2. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 0], Q=[-12, -2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-11, -2, 125)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-11, -2, 125).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
3. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, -1], Q=[-10, 2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-10, 3, 109)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-10, 3, 109).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
4. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 1], Q=[-8, -3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-9, -4, 97)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-9, -4, 97).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
5. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 0], Q=[-10, 5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-8, 5, 89)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-8, 5, 89).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
6. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, -1], Q=[-8, -2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-7, -1, 50)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-7, -1, 50).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
7. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 1], Q=[-6, 3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-6, 2, 40)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-6, 2, 40).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
8. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 0], Q=[-4, -3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-5, -3, 34)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-5, -3, 34).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
9. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, -1], Q=[-6, 3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-4, 4, 32)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-4, 4, 32).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
10. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 1], Q=[-4, -4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-3, -5, 34)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-3, -5, 34).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
11. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[-2, 1]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-2, 1, 5)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-2, 1, 5).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
12. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, -1], Q=[0, -3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(-1, -2, 5)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (-1, -2, 5).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
13. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 1], Q=[-2, 4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(0, 3, 9)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (0, 3, 9).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
14. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 0], Q=[0, -4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(1, -4, 17)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (1, -4, 17).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
15. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, -1], Q=[2, 4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(2, 5, 29)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (2, 5, 29).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
16. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 1], Q=[4, 0]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(3, -1, 10)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (3, -1, 10).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
17. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 0], Q=[2, 2]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(4, 2, 20)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (4, 2, 20).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
18. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, -1], Q=[4, -4]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(5, -3, 34)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (5, -3, 34).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
19. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 1], Q=[6, 5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(6, 4, 52)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (6, 4, 52).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
20. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 0], Q=[8, -5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(7, -5, 74)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (7, -5, 74).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
21. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, -1], Q=[6, 0]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(8, 1, 65)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (8, 1, 65).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
22. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 1], Q=[8, -1]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(9, -2, 85)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (9, -2, 85).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
23. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[10, 3]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(10, 3, 109)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (10, 3, 109).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
24. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, -1], Q=[12, -5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(11, -4, 137)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (11, -4, 137).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
25. Directed Components and Squared Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 1], Q=[10, 6]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
View answer and explanation
(12, 5, 169)
Coordinate differences are directed while their squared sum is nonnegative; the actual distance is its nonnegative principal root, not a signed component. The exact result, retaining the requested order, is (12, 5, 169).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-displacement-square · candidate
26. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [0, 0], [0, 0]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(0, 0)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
27. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [3, 4], [3, 4]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(5, 0)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (5, 0).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
28. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [0, 0], [5, 12]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(0, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (0, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
29. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [3, 4], [8, 16]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(5, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (5, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
30. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [6, 8], [11, 20]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(10, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (10, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
31. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [9, 12], [14, 24]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(15, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (15, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
32. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [12, 16], [17, 28]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(20, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (20, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
33. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [15, 20], [20, 32]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(25, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (25, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
34. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [18, 24], [23, 36]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(30, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (30, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
35. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [21, 28], [26, 40]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(35, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (35, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
36. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [24, 32], [29, 44]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(40, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (40, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
37. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [27, 36], [32, 48]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(45, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (45, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
38. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [30, 40], [35, 52]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(50, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (50, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
39. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [33, 44], [38, 56]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(55, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (55, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
40. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [36, 48], [41, 60]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(60, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (60, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
41. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [39, 52], [44, 64]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(65, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (65, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
42. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [42, 56], [47, 68]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(70, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (70, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
43. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [45, 60], [50, 72]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(75, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (75, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
44. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [48, 64], [53, 76]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(80, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (80, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
45. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [51, 68], [56, 80]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(85, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (85, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
46. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [54, 72], [59, 84]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(90, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (90, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
47. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [57, 76], [62, 88]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(95, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (95, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
48. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [60, 80], [65, 92]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(100, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (100, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
49. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [63, 84], [68, 96]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(105, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (105, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
50. Ordered Integer Principal Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [66, 88], [71, 100]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.
View answer and explanation
(110, 13)
Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (110, 13).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-integer-distance-pair · candidate
51. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[0, 0], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, 0/1, 0/1, 0/1)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (0/1, 0/1, 0/1, 0/1).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
52. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, 0], Q=[1, 0], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1/4, 1/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 0/1, 1/4, 1/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
53. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, 1], Q=[3, 4], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 5/2, 17/2, 17/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 5/2, 17/2, 17/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
54. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-3, 1], Q=[4, 5], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 3/1, 65/4, 65/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 3/1, 65/4, 65/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
55. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-4, 1], Q=[5, 6], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 7/2, 53/2, 53/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 7/2, 53/2, 53/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
56. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-5, 1], Q=[6, 7], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 4/1, 157/4, 157/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 4/1, 157/4, 157/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
57. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-6, 1], Q=[7, 8], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 9/2, 109/2, 109/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 9/2, 109/2, 109/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
58. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-7, 1], Q=[8, 9], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 5/1, 289/4, 289/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 5/1, 289/4, 289/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
59. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-8, 1], Q=[9, 10], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 11/2, 185/2, 185/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 11/2, 185/2, 185/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
60. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-9, 1], Q=[10, 11], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 6/1, 461/4, 461/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 6/1, 461/4, 461/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
61. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-10, 1], Q=[11, 12], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 13/2, 281/2, 281/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 13/2, 281/2, 281/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
62. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-11, 1], Q=[12, 13], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 7/1, 673/4, 673/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 7/1, 673/4, 673/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
63. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-12, 1], Q=[13, 14], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 15/2, 397/2, 397/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 15/2, 397/2, 397/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
64. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-13, 1], Q=[14, 15], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 8/1, 925/4, 925/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 8/1, 925/4, 925/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
65. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-14, 1], Q=[15, 16], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 17/2, 533/2, 533/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 17/2, 533/2, 533/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
66. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-15, 1], Q=[16, 17], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 9/1, 1217/4, 1217/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 9/1, 1217/4, 1217/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
67. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-16, 1], Q=[17, 18], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 19/2, 689/2, 689/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 19/2, 689/2, 689/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
68. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-17, 1], Q=[18, 19], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 10/1, 1549/4, 1549/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 10/1, 1549/4, 1549/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
69. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-18, 1], Q=[19, 20], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 21/2, 865/2, 865/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 21/2, 865/2, 865/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
70. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-19, 1], Q=[20, 21], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 11/1, 1921/4, 1921/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 11/1, 1921/4, 1921/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
71. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-20, 1], Q=[21, 22], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 23/2, 1061/2, 1061/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 23/2, 1061/2, 1061/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
72. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-21, 1], Q=[22, 23], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 12/1, 2333/4, 2333/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 12/1, 2333/4, 2333/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
73. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-22, 1], Q=[23, 24], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 25/2, 1277/2, 1277/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 25/2, 1277/2, 1277/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
74. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-23, 1], Q=[24, 25], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 13/1, 2785/4, 2785/4)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 13/1, 2785/4, 2785/4).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
75. Exact Midpoint and Two Squared Segment Lengths
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-24, 1], Q=[25, 26], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 27/2, 1513/2, 1513/2)
Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 27/2, 1513/2, 1513/2).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-midpoint-equidistance · candidate
76. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(0,0), R=(0,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(0, 0, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (0, 0, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
77. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(1,0), R=(2,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(2, 2, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (2, 2, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
78. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-1,0), R=(2,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(4, 2, 2, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (4, 2, 2, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
79. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-3,0), R=(3,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(9, 3, 6, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (9, 3, 6, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
80. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(2,0), R=(4,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(4, 4, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (4, 4, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
81. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-5,0), R=(5,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(15, 5, 10, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (15, 5, 10, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
82. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(3,0), R=(6,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(6, 6, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (6, 6, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
83. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-7,0), R=(7,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(21, 7, 14, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (21, 7, 14, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
84. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(4,0), R=(8,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(8, 8, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (8, 8, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
85. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-9,0), R=(9,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(27, 9, 18, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (27, 9, 18, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
86. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(5,0), R=(10,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(10, 10, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (10, 10, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
87. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-11,0), R=(11,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(33, 11, 22, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (33, 11, 22, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
88. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(6,0), R=(12,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(12, 12, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (12, 12, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
89. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-13,0), R=(13,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(39, 13, 26, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (39, 13, 26, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
90. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(7,0), R=(14,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(14, 14, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (14, 14, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
91. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-15,0), R=(15,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(45, 15, 30, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (45, 15, 30, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
92. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(8,0), R=(16,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(16, 16, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (16, 16, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
93. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-17,0), R=(17,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(51, 17, 34, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (51, 17, 34, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
94. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(9,0), R=(18,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(18, 18, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (18, 18, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
95. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-19,0), R=(19,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(57, 19, 38, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (57, 19, 38, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
96. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(10,0), R=(20,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(20, 20, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (20, 20, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
97. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-21,0), R=(21,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(63, 21, 42, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (63, 21, 42, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
98. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(11,0), R=(22,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(22, 22, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (22, 22, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
99. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-23,0), R=(23,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(69, 23, 46, No)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (69, 23, 46, No).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
100. Axis Path Sum versus Endpoint Length
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(12,0), R=(24,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
View answer and explanation
(24, 24, 0, Yes)
Use absolute differences for nonnegative axis segment lengths; the path sum is not the straight-line endpoint length, and Q outside the segment may yield a positive defect. The exact result, retaining the requested order, is (24, 24, 0, Yes).
Reviewed step-by-step derivation
Euclidean length · qf-density12-length-triangle-defect · candidate
Canonical question JSONL · English JSONL · Validation report