# Plane Coordinate Isometry and Uniform Scaling Foundations

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-20261006-advance-20261007-r07-geometric-transformations-foundations · 2026-10-06 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(0, 0) and translation vector t=[0, 0]. Find the coordinates of P+t.

View answer and explanation(0, 0)

Add the two components of the specified translation, obtaining the exact image (0, 0). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0001/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 2. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-11, 0) and translation vector t=[0, 0]. Find the coordinates of P+t.

View answer and explanation(-11, 0)

Add the two components of the specified translation, obtaining the exact image (-11, 0). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0002/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 3. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-10, -1) and translation vector t=[1, 2]. Find the coordinates of P+t.

View answer and explanation(-9, 1)

Add the two components of the specified translation, obtaining the exact image (-9, 1). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0003/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 4. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-9, -2) and translation vector t=[-1, -1]. Find the coordinates of P+t.

View answer and explanation(-10, -3)

Add the two components of the specified translation, obtaining the exact image (-10, -3). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0004/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 5. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-8, -3) and translation vector t=[0, 1]. Find the coordinates of P+t.

View answer and explanation(-8, -2)

Add the two components of the specified translation, obtaining the exact image (-8, -2). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0005/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 6. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-7, -4) and translation vector t=[1, -2]. Find the coordinates of P+t.

View answer and explanation(-6, -6)

Add the two components of the specified translation, obtaining the exact image (-6, -6). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0006/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 7. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-6, -5) and translation vector t=[-1, 0]. Find the coordinates of P+t.

View answer and explanation(-7, -5)

Add the two components of the specified translation, obtaining the exact image (-7, -5). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0007/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 8. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-5, -6) and translation vector t=[0, 2]. Find the coordinates of P+t.

View answer and explanation(-5, -4)

Add the two components of the specified translation, obtaining the exact image (-5, -4). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0008/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 9. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-4, -7) and translation vector t=[1, -1]. Find the coordinates of P+t.

View answer and explanation(-3, -8)

Add the two components of the specified translation, obtaining the exact image (-3, -8). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0009/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 10. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-3, -8) and translation vector t=[-1, 1]. Find the coordinates of P+t.

View answer and explanation(-4, -7)

Add the two components of the specified translation, obtaining the exact image (-4, -7). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0010/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 11. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-2, -9) and translation vector t=[0, -2]. Find the coordinates of P+t.

View answer and explanation(-2, -11)

Add the two components of the specified translation, obtaining the exact image (-2, -11). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0011/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 12. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(-1, -10) and translation vector t=[1, 0]. Find the coordinates of P+t.

View answer and explanation(0, -10)

Add the two components of the specified translation, obtaining the exact image (0, -10). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0012/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 13. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(0, -11) and translation vector t=[-1, 2]. Find the coordinates of P+t.

View answer and explanation(-1, -9)

Add the two components of the specified translation, obtaining the exact image (-1, -9). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0013/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 14. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(1, -12) and translation vector t=[0, -1]. Find the coordinates of P+t.

View answer and explanation(1, -13)

Add the two components of the specified translation, obtaining the exact image (1, -13). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0014/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 15. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(2, -13) and translation vector t=[1, 1]. Find the coordinates of P+t.

View answer and explanation(3, -12)

Add the two components of the specified translation, obtaining the exact image (3, -12). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0015/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 16. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(3, -14) and translation vector t=[-1, -2]. Find the coordinates of P+t.

View answer and explanation(2, -16)

Add the two components of the specified translation, obtaining the exact image (2, -16). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0016/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 17. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(4, -15) and translation vector t=[0, 0]. Find the coordinates of P+t.

View answer and explanation(4, -15)

Add the two components of the specified translation, obtaining the exact image (4, -15). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0017/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 18. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(5, -16) and translation vector t=[1, 2]. Find the coordinates of P+t.

View answer and explanation(6, -14)

Add the two components of the specified translation, obtaining the exact image (6, -14). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0018/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 19. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(6, -17) and translation vector t=[-1, -1]. Find the coordinates of P+t.

View answer and explanation(5, -18)

Add the two components of the specified translation, obtaining the exact image (5, -18). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0019/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 20. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(7, -18) and translation vector t=[0, 1]. Find the coordinates of P+t.

View answer and explanation(7, -17)

Add the two components of the specified translation, obtaining the exact image (7, -17). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0020/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 21. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(8, -19) and translation vector t=[1, -2]. Find the coordinates of P+t.

View answer and explanation(9, -21)

Add the two components of the specified translation, obtaining the exact image (9, -21). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0021/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 22. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(9, -20) and translation vector t=[-1, 0]. Find the coordinates of P+t.

View answer and explanation(8, -20)

Add the two components of the specified translation, obtaining the exact image (8, -20). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0022/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 23. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(10, -21) and translation vector t=[0, 2]. Find the coordinates of P+t.

View answer and explanation(10, -19)

Add the two components of the specified translation, obtaining the exact image (10, -19). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0023/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 24. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(11, -22) and translation vector t=[1, -1]. Find the coordinates of P+t.

View answer and explanation(12, -23)

Add the two components of the specified translation, obtaining the exact image (12, -23). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0024/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 25. Exact Image Point under Translation

Use one equal-unit orthonormal Euclidean Cartesian frame. The point is P=(12, -23) and translation vector t=[-1, 1]. Find the coordinates of P+t.

View answer and explanation(11, -22)

Add the two components of the specified translation, obtaining the exact image (11, -22). Retain the actual image position rather than only a difference between points.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0025/)

Plane-coordinate isometries and uniform scaling · qf-transform-translation-point · candidate





## 26. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(0, 0) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(0, 0)

Each quarter-turn sends (x,y) to (−y,x), giving (0, 0) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0026/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 27. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-11, 2) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-2, -11)

Each quarter-turn sends (x,y) to (−y,x), giving (-2, -11) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0027/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 28. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-10, 3) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(10, -3)

Each quarter-turn sends (x,y) to (−y,x), giving (10, -3) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0028/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 29. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-9, 4) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(4, 9)

Each quarter-turn sends (x,y) to (−y,x), giving (4, 9) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0029/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 30. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-8, 5) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-8, 5)

Each quarter-turn sends (x,y) to (−y,x), giving (-8, 5) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0030/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 31. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-7, 6) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-6, -7)

Each quarter-turn sends (x,y) to (−y,x), giving (-6, -7) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0031/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 32. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-6, 7) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(6, -7)

Each quarter-turn sends (x,y) to (−y,x), giving (6, -7) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0032/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 33. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-5, 8) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(8, 5)

Each quarter-turn sends (x,y) to (−y,x), giving (8, 5) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0033/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 34. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-4, 9) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-4, 9)

Each quarter-turn sends (x,y) to (−y,x), giving (-4, 9) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0034/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 35. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-3, 10) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-10, -3)

Each quarter-turn sends (x,y) to (−y,x), giving (-10, -3) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0035/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 36. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-2, 11) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(2, -11)

Each quarter-turn sends (x,y) to (−y,x), giving (2, -11) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0036/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 37. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(-1, 12) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(12, 1)

Each quarter-turn sends (x,y) to (−y,x), giving (12, 1) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0037/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 38. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(0, 13) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(0, 13)

Each quarter-turn sends (x,y) to (−y,x), giving (0, 13) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0038/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 39. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(1, 14) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-14, 1)

Each quarter-turn sends (x,y) to (−y,x), giving (-14, 1) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0039/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 40. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(2, 15) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-2, -15)

Each quarter-turn sends (x,y) to (−y,x), giving (-2, -15) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0040/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 41. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(3, 16) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(16, -3)

Each quarter-turn sends (x,y) to (−y,x), giving (16, -3) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0041/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 42. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(4, 17) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(4, 17)

Each quarter-turn sends (x,y) to (−y,x), giving (4, 17) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0042/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 43. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(5, 18) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-18, 5)

Each quarter-turn sends (x,y) to (−y,x), giving (-18, 5) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0043/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 44. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(6, 19) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-6, -19)

Each quarter-turn sends (x,y) to (−y,x), giving (-6, -19) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0044/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 45. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(7, 20) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(20, -7)

Each quarter-turn sends (x,y) to (−y,x), giving (20, -7) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0045/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 46. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(8, 21) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(8, 21)

Each quarter-turn sends (x,y) to (−y,x), giving (8, 21) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0046/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 47. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(9, 22) about the origin counterclockwise by 1 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-22, 9)

Each quarter-turn sends (x,y) to (−y,x), giving (-22, 9) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0047/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 48. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(10, 23) about the origin counterclockwise by 2 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(-10, -23)

Each quarter-turn sends (x,y) to (−y,x), giving (-10, -23) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0048/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 49. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(11, 24) about the origin counterclockwise by 3 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(24, -11)

Each quarter-turn sends (x,y) to (−y,x), giving (24, -11) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0049/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 50. Counterclockwise Quarter-Turns about the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Rotate P=(12, 25) about the origin counterclockwise by 0 quarter-turns of 90°. Find the image coordinates.

View answer and explanation(12, 25)

Each quarter-turn sends (x,y) to (−y,x), giving (12, 25) after the stated number of turns. Center and direction are explicit; this one image calculation is not by itself a proof of a whole-plane property.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0050/)

Plane-coordinate isometries and uniform scaling · qf-transform-quarter-turn-point · candidate





## 51. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(0, 0) across the x-axis. Find the image coordinates.

View answer and explanation(0, 0)

Use the specified reflection formula (x,y)→(x,−y), giving (0, 0). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0051/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 52. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-11, 3) across the y-axis. Find the image coordinates.

View answer and explanation(11, 3)

Use the specified reflection formula (x,y)→(−x,y), giving (11, 3). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0052/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 53. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-10, 4) across the line y=x. Find the image coordinates.

View answer and explanation(4, -10)

Use the specified reflection formula (x,y)→(y,x), giving (4, -10). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0053/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 54. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-9, 5) across the line y=−x. Find the image coordinates.

View answer and explanation(-5, 9)

Use the specified reflection formula (x,y)→(−y,−x), giving (-5, 9). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0054/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 55. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-8, 6) across the x-axis. Find the image coordinates.

View answer and explanation(-8, -6)

Use the specified reflection formula (x,y)→(x,−y), giving (-8, -6). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0055/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 56. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-7, 7) across the y-axis. Find the image coordinates.

View answer and explanation(7, 7)

Use the specified reflection formula (x,y)→(−x,y), giving (7, 7). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0056/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 57. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-6, 8) across the line y=x. Find the image coordinates.

View answer and explanation(8, -6)

Use the specified reflection formula (x,y)→(y,x), giving (8, -6). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0057/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 58. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-5, 9) across the line y=−x. Find the image coordinates.

View answer and explanation(-9, 5)

Use the specified reflection formula (x,y)→(−y,−x), giving (-9, 5). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0058/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 59. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-4, 10) across the x-axis. Find the image coordinates.

View answer and explanation(-4, -10)

Use the specified reflection formula (x,y)→(x,−y), giving (-4, -10). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0059/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 60. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-3, 11) across the y-axis. Find the image coordinates.

View answer and explanation(3, 11)

Use the specified reflection formula (x,y)→(−x,y), giving (3, 11). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0060/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 61. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-2, 12) across the line y=x. Find the image coordinates.

View answer and explanation(12, -2)

Use the specified reflection formula (x,y)→(y,x), giving (12, -2). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0061/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 62. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(-1, 13) across the line y=−x. Find the image coordinates.

View answer and explanation(-13, 1)

Use the specified reflection formula (x,y)→(−y,−x), giving (-13, 1). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0062/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 63. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(0, 14) across the x-axis. Find the image coordinates.

View answer and explanation(0, -14)

Use the specified reflection formula (x,y)→(x,−y), giving (0, -14). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0063/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 64. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(1, 15) across the y-axis. Find the image coordinates.

View answer and explanation(-1, 15)

Use the specified reflection formula (x,y)→(−x,y), giving (-1, 15). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0064/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 65. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(2, 16) across the line y=x. Find the image coordinates.

View answer and explanation(16, 2)

Use the specified reflection formula (x,y)→(y,x), giving (16, 2). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0065/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 66. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(3, 17) across the line y=−x. Find the image coordinates.

View answer and explanation(-17, -3)

Use the specified reflection formula (x,y)→(−y,−x), giving (-17, -3). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0066/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 67. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(4, 18) across the x-axis. Find the image coordinates.

View answer and explanation(4, -18)

Use the specified reflection formula (x,y)→(x,−y), giving (4, -18). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0067/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 68. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(5, 19) across the y-axis. Find the image coordinates.

View answer and explanation(-5, 19)

Use the specified reflection formula (x,y)→(−x,y), giving (-5, 19). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0068/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 69. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(6, 20) across the line y=x. Find the image coordinates.

View answer and explanation(20, 6)

Use the specified reflection formula (x,y)→(y,x), giving (20, 6). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0069/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 70. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(7, 21) across the line y=−x. Find the image coordinates.

View answer and explanation(-21, -7)

Use the specified reflection formula (x,y)→(−y,−x), giving (-21, -7). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0070/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 71. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(8, 22) across the x-axis. Find the image coordinates.

View answer and explanation(8, -22)

Use the specified reflection formula (x,y)→(x,−y), giving (8, -22). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0071/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 72. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(9, 23) across the y-axis. Find the image coordinates.

View answer and explanation(-9, 23)

Use the specified reflection formula (x,y)→(−x,y), giving (-9, 23). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0072/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 73. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(10, 24) across the line y=x. Find the image coordinates.

View answer and explanation(24, 10)

Use the specified reflection formula (x,y)→(y,x), giving (24, 10). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0073/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 74. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(11, 25) across the line y=−x. Find the image coordinates.

View answer and explanation(-25, -11)

Use the specified reflection formula (x,y)→(−y,−x), giving (-25, -11). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0074/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 75. Reflection across a Specified Line through the Origin

Use one equal-unit orthonormal Euclidean Cartesian frame. Reflect P=(12, 26) across the x-axis. Find the image coordinates.

View answer and explanation(12, -26)

Use the specified reflection formula (x,y)→(x,−y), giving (12, -26). A line reflection is not central inversion (−x,−y).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0075/)

Plane-coordinate isometries and uniform scaling · qf-transform-axis-reflection-point · candidate





## 76. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(0)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationdegenerate constant map

Classify the global formula by |c|, giving degenerate constant map: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0076/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 77. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-1)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationwhole-plane isometry

Classify the global formula by |c|, giving whole-plane isometry: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0077/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 78. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(2)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=2

Classify the global formula by |c|, giving non-isometric similarity; scale=2: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0078/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 79. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-3)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=3

Classify the global formula by |c|, giving non-isometric similarity; scale=3: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0079/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 80. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(4)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=4

Classify the global formula by |c|, giving non-isometric similarity; scale=4: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0080/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 81. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-5)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=5

Classify the global formula by |c|, giving non-isometric similarity; scale=5: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0081/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 82. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(6)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=6

Classify the global formula by |c|, giving non-isometric similarity; scale=6: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0082/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 83. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-7)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=7

Classify the global formula by |c|, giving non-isometric similarity; scale=7: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0083/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 84. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(8)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=8

Classify the global formula by |c|, giving non-isometric similarity; scale=8: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0084/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 85. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-9)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=9

Classify the global formula by |c|, giving non-isometric similarity; scale=9: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0085/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 86. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(10)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=10

Classify the global formula by |c|, giving non-isometric similarity; scale=10: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0086/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 87. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-11)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=11

Classify the global formula by |c|, giving non-isometric similarity; scale=11: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0087/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 88. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(12)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=12

Classify the global formula by |c|, giving non-isometric similarity; scale=12: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0088/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 89. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-13)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=13

Classify the global formula by |c|, giving non-isometric similarity; scale=13: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0089/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 90. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(14)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=14

Classify the global formula by |c|, giving non-isometric similarity; scale=14: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0090/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 91. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-15)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=15

Classify the global formula by |c|, giving non-isometric similarity; scale=15: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0091/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 92. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(16)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=16

Classify the global formula by |c|, giving non-isometric similarity; scale=16: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0092/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 93. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-17)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=17

Classify the global formula by |c|, giving non-isometric similarity; scale=17: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0093/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 94. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(18)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=18

Classify the global formula by |c|, giving non-isometric similarity; scale=18: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0094/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 95. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-19)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=19

Classify the global formula by |c|, giving non-isometric similarity; scale=19: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0095/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 96. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(20)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=20

Classify the global formula by |c|, giving non-isometric similarity; scale=20: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0096/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 97. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-21)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=21

Classify the global formula by |c|, giving non-isometric similarity; scale=21: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0097/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 98. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(22)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=22

Classify the global formula by |c|, giving non-isometric similarity; scale=22: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0098/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 99. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(-23)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=23

Classify the global formula by |c|, giving non-isometric similarity; scale=23: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0099/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate





## 100. Whole-Plane Classification of Uniform Scaling

Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(24)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.

View answer and explanationnon-isometric similarity; scale=24

Classify the global formula by |c|, giving non-isometric similarity; scale=24: c=0 is degenerate, |c|=1 is isometric, and other nonzero c have positive length scale |c|. In particular c=−1 is a 180° origin rotation, not a line reflection. One fixed point or coincident pair cannot establish global isometry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r07-geometric-transformations-0100/)

Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r07-geometric-transformations-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261006-advance-20261007-r07-geometric-transformations-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r07-geometric-transformations-foundations/validation-report.json)