# 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r06-euclidean-length-0100/)

Euclidean length · qf-density12-length-triangle-defect · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r06-euclidean-length-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261012-r06-euclidean-length-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r06-euclidean-length-foundations/validation-report.json)