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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
degenerate 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
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 explanation
whole-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
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 explanation
non-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
Plane-coordinate isometries and uniform scaling · qf-transform-uniform-scale-class · candidate
Canonical question JSONL · English JSONL · Validation report