# Equal-Unit Orthonormal Euclidean Distance Foundations

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261005-advance-20261006-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.



## 1. Squared Euclidean Distance

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.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0001/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 2. Squared Euclidean Distance

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.

View answer and explanation5

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0002/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 3. Squared Euclidean Distance

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.

View answer and explanation13

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0003/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 4. Squared Euclidean Distance

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.

View answer and explanation25

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0004/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 5. Squared Euclidean Distance

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.

View answer and explanation16

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0005/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 6. Squared Euclidean Distance

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.

View answer and explanation61

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0006/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 7. Squared Euclidean Distance

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.

View answer and explanation85

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0007/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 8. Squared Euclidean Distance

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.

View answer and explanation113

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0008/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 9. Squared Euclidean Distance

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.

View answer and explanation64

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0009/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 10. Squared Euclidean Distance

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.

View answer and explanation181

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0010/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 11. Squared Euclidean Distance

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.

View answer and explanation221

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0011/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 12. Squared Euclidean Distance

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.

View answer and explanation265

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0012/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 13. Squared Euclidean Distance

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.

View answer and explanation144

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0013/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 14. Squared Euclidean Distance

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.

View answer and explanation365

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0014/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 15. Squared Euclidean Distance

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.

View answer and explanation421

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0015/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 16. Squared Euclidean Distance

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.

View answer and explanation481

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0016/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 17. Squared Euclidean Distance

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.

View answer and explanation256

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0017/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 18. Squared Euclidean Distance

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.

View answer and explanation613

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0018/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 19. Squared Euclidean Distance

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.

View answer and explanation685

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0019/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 20. Squared Euclidean Distance

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.

View answer and explanation761

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0020/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 21. Squared Euclidean Distance

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.

View answer and explanation400

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0021/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 22. Squared Euclidean Distance

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.

View answer and explanation925

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0022/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 23. Squared Euclidean Distance

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.

View answer and explanation1013

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0023/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 24. Squared Euclidean Distance

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.

View answer and explanation1105

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0024/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 25. Squared Euclidean Distance

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.

View answer and explanation576

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0025/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-squared · candidate





## 26. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0026/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 27. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation10

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0027/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 28. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation3

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0028/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 29. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation26

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0029/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 30. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation5

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0030/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 31. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation50

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0031/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 32. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation7

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0032/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 33. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation82

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0033/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 34. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation9

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0034/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 35. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation122

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0035/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 36. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation11

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0036/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 37. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation170

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0037/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 38. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation13

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0038/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 39. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation226

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0039/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 40. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation15

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0040/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 41. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation290

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0041/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 42. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation17

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0042/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 43. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation362

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0043/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 44. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation19

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0044/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 45. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation442

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0045/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 46. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation21

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0046/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 47. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation530

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0047/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 48. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation23

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0048/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 49. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation626

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0049/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 50. Nonnegative Distance in a Perfect-Square Slice

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.

View answer and explanation25

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0050/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-perfect-square · candidate





## 51. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0051/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 52. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0052/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 53. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0053/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 54. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0054/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 55. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0055/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 56. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0056/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 57. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0057/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 58. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0058/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 59. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0059/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 60. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0060/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 61. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0061/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 62. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0062/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 63. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0063/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 64. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0064/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 65. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0065/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 66. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0066/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 67. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0067/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 68. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0068/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 69. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0069/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 70. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0070/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 71. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0071/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 72. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0072/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 73. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0073/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 74. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0074/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 75. Sign and Square Eligibility of a Candidate Distance

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.

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0075/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-candidate · candidate





## 76. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 0 and 0 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0076/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 77. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 5 and 17 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0077/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 78. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 13 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0078/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 79. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 25 and 25 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0079/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 80. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 16 and 36 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0080/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 81. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 61 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0081/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 82. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 85 and 85 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0082/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 83. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 113 and 149 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0083/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 84. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 64 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0084/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 85. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 181 and 181 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0085/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 86. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 221 and 269 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0086/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 87. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 265 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0087/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 88. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 144 and 144 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0088/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 89. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 365 and 425 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0089/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 90. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 421 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0090/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 91. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 481 and 481 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0091/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 92. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 256 and 324 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0092/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 93. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 613 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0093/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 94. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 685 and 685 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0094/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 95. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 761 and 845 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0095/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 96. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 400 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0096/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 97. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 925 and 925 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0097/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 98. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation<

Both distances are nonnegative, so compare their squares 1013 and 1109 exactly. The relation is <, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0098/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 99. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation>

Both distances are nonnegative, so compare their squares 1105 and 0 exactly. The relation is >, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0099/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate





## 100. Exact Comparison of Two Euclidean Distances

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.

View answer and explanation=

Both distances are nonnegative, so compare their squares 576 and 576 exactly. The relation is =, without approximating square roots.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r04-distance-formula-0100/)

Euclidean distance in an equal-unit orthonormal frame · qf-distance-comparison · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r04-distance-formula-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261005-advance-20261006-r04-distance-formula-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r04-distance-formula-foundations/validation-report.json)