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-20261005-advance-20261006-r04-distance-formula-foundations · 2026-10-05 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). Find d(P,Q)² for the Euclidean L2 distance.
0
Add the squares of the two coordinate differences to obtain 0. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
5
Add the squares of the two coordinate differences to obtain 5. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-8, 6). Find d(P,Q)² for the Euclidean L2 distance.
13
Add the squares of the two coordinate differences to obtain 13. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
25
Add the squares of the two coordinate differences to obtain 25. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 5). Find d(P,Q)² for the Euclidean L2 distance.
16
Add the squares of the two coordinate differences to obtain 16. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
61
Add the squares of the two coordinate differences to obtain 61. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(0, 6). Find d(P,Q)² for the Euclidean L2 distance.
85
Add the squares of the two coordinate differences to obtain 85. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
113
Add the squares of the two coordinate differences to obtain 113. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 5). Find d(P,Q)² for the Euclidean L2 distance.
64
Add the squares of the two coordinate differences to obtain 64. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
181
Add the squares of the two coordinate differences to obtain 181. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(8, 6). Find d(P,Q)² for the Euclidean L2 distance.
221
Add the squares of the two coordinate differences to obtain 221. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
265
Add the squares of the two coordinate differences to obtain 265. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 5). Find d(P,Q)² for the Euclidean L2 distance.
144
Add the squares of the two coordinate differences to obtain 144. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
365
Add the squares of the two coordinate differences to obtain 365. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(16, 6). Find d(P,Q)² for the Euclidean L2 distance.
421
Add the squares of the two coordinate differences to obtain 421. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
481
Add the squares of the two coordinate differences to obtain 481. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 5). Find d(P,Q)² for the Euclidean L2 distance.
256
Add the squares of the two coordinate differences to obtain 256. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
613
Add the squares of the two coordinate differences to obtain 613. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(24, 6). Find d(P,Q)² for the Euclidean L2 distance.
685
Add the squares of the two coordinate differences to obtain 685. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
761
Add the squares of the two coordinate differences to obtain 761. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 5). Find d(P,Q)² for the Euclidean L2 distance.
400
Add the squares of the two coordinate differences to obtain 400. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
925
Add the squares of the two coordinate differences to obtain 925. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(32, 6). Find d(P,Q)² for the Euclidean L2 distance.
1013
Add the squares of the two coordinate differences to obtain 1013. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.
1105
Add the squares of the two coordinate differences to obtain 1105. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 5). Find d(P,Q)² for the Euclidean L2 distance.
576
Add the squares of the two coordinate differences to obtain 576. This is the squared distance, not its root; coincidence legitimately gives zero.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
0
The squared distance is 0, and its nonnegative principal root is 0. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-19, 10). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
10
The squared distance is 100, and its nonnegative principal root is 10. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-10, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
3
The squared distance is 9, and its nonnegative principal root is 3. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-33, 12). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
26
The squared distance is 676, and its nonnegative principal root is 26. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
5
The squared distance is 25, and its nonnegative principal root is 5. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-55, 14). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
50
The squared distance is 2500, and its nonnegative principal root is 50. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(-6, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
7
The squared distance is 49, and its nonnegative principal root is 7. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-85, 16). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
82
The squared distance is 6724, and its nonnegative principal root is 82. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
9
The squared distance is 81, and its nonnegative principal root is 9. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-123, 18). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
122
The squared distance is 14884, and its nonnegative principal root is 122. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(-2, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
11
The squared distance is 121, and its nonnegative principal root is 11. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-169, 20). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
170
The squared distance is 28900, and its nonnegative principal root is 170. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
13
The squared distance is 169, and its nonnegative principal root is 13. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-223, 22). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
226
The squared distance is 51076, and its nonnegative principal root is 226. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(2, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
15
The squared distance is 225, and its nonnegative principal root is 15. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-285, 24). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
290
The squared distance is 84100, and its nonnegative principal root is 290. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
17
The squared distance is 289, and its nonnegative principal root is 17. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-355, 26). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
362
The squared distance is 131044, and its nonnegative principal root is 362. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(6, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
19
The squared distance is 361, and its nonnegative principal root is 19. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-433, 28). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
442
The squared distance is 195364, and its nonnegative principal root is 442. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
21
The squared distance is 441, and its nonnegative principal root is 21. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-519, 30). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
530
The squared distance is 280900, and its nonnegative principal root is 530. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(10, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
23
The squared distance is 529, and its nonnegative principal root is 23. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-613, 32). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
626
The squared distance is 391876, and its nonnegative principal root is 626. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 6). The sum of squared differences is a perfect square. Find the nonnegative Euclidean L2 distance exactly.
25
The squared distance is 625, and its nonnegative principal root is 25. A negative root is not a distance, and no floating approximation is used.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). Is the candidate d=0 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 0. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-19, 10). Is the candidate d=-10 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 100. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-10, 6). Is the candidate d=4 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 9. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-33, 12). Is the candidate d=26 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 676. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 6). Is the candidate d=5 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 25. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-55, 14). Is the candidate d=-50 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 2500. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(-6, 6). Is the candidate d=8 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 49. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-85, 16). Is the candidate d=82 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 6724. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 6). Is the candidate d=9 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 81. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-123, 18). Is the candidate d=-122 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 14884. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(-2, 6). Is the candidate d=12 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 121. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-169, 20). Is the candidate d=170 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 28900. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 6). Is the candidate d=13 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 169. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-223, 22). Is the candidate d=-226 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 51076. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(2, 6). Is the candidate d=16 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 225. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-285, 24). Is the candidate d=290 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 84100. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 6). Is the candidate d=17 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 289. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-355, 26). Is the candidate d=-362 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 131044. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(6, 6). Is the candidate d=20 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 361. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-433, 28). Is the candidate d=442 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 195364. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 6). Is the candidate d=21 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 441. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-519, 30). Is the candidate d=-530 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 280900. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(10, 6). Is the candidate d=24 the Euclidean L2 distance? Check both its sign and its square.
No
The squared distance is 529. Require d≥0 and d² equal to that value, giving No; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-613, 32). Is the candidate d=626 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 391876. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 6). Is the candidate d=25 the Euclidean L2 distance? Check both its sign and its square.
Yes
The squared distance is 625. Require d≥0 and d² equal to that value, giving Yes; a negative candidate fails even when its square agrees.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-12, 5) and Q=(-12, 5). In the same frame, U=(5, -3) and V=(5, -3). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 0 and 0 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-11, 4) and Q=(-12, 6). In the same frame, U=(4, -4) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 5 and 17 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-10, 3) and Q=(-8, 6). In the same frame, U=(3, -5) and V=(3, -5). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 13 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-9, 2) and Q=(-12, 6). In the same frame, U=(2, -6) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 25 and 25 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-8, 1) and Q=(-8, 5). In the same frame, U=(1, -7) and V=(1, -1). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 16 and 36 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-7, 0) and Q=(-12, 6). In the same frame, U=(0, -8) and V=(0, -8). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 61 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-6, -1) and Q=(0, 6). In the same frame, U=(-1, -9) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 85 and 85 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-5, -2) and Q=(-12, 6). In the same frame, U=(-2, -10) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 113 and 149 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-4, -3) and Q=(-4, 5). In the same frame, U=(-3, -11) and V=(-3, -11). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 64 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-3, -4) and Q=(-12, 6). In the same frame, U=(-4, -12) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 181 and 181 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-2, -5) and Q=(8, 6). In the same frame, U=(-5, -13) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 221 and 269 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(-1, -6) and Q=(-12, 6). In the same frame, U=(-6, -14) and V=(-6, -14). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 265 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(0, -7) and Q=(0, 5). In the same frame, U=(-7, -15) and V=(-7, -3). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 144 and 144 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-12, 6). In the same frame, U=(-8, -16) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 365 and 425 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(2, -9) and Q=(16, 6). In the same frame, U=(-9, -17) and V=(-9, -17). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 421 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-12, 6). In the same frame, U=(-10, -18) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 481 and 481 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 5). In the same frame, U=(-11, -19) and V=(-11, -1). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 256 and 324 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(5, -12) and Q=(-12, 6). In the same frame, U=(-12, -20) and V=(-12, -20). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 613 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(6, -13) and Q=(24, 6). In the same frame, U=(-13, -21) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 685 and 685 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(7, -14) and Q=(-12, 6). In the same frame, U=(-14, -22) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 761 and 845 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(8, -15) and Q=(8, 5). In the same frame, U=(-15, -23) and V=(-15, -23). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 400 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(9, -16) and Q=(-12, 6). In the same frame, U=(-16, -24) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 925 and 925 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(10, -17) and Q=(32, 6). In the same frame, U=(-17, -25) and V=(5, 0). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
<
Both distances are nonnegative, so compare their squares 1013 and 1109 exactly. The relation is <, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(11, -18) and Q=(-12, 6). In the same frame, U=(-18, -26) and V=(-18, -26). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
>
Both distances are nonnegative, so compare their squares 1105 and 0 exactly. The relation is >, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
Use one equal-unit orthonormal Euclidean Cartesian frame. P=(12, -19) and Q=(12, 5). In the same frame, U=(-19, -27) and V=(-19, -3). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.
=
Both distances are nonnegative, so compare their squares 576 and 576 exactly. The relation is =, without approximating square roots.
Reviewed step-by-step derivation
Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate
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