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Sequence of Rigid Motions to Show Congruence

Sequence of Rigid Motions to Show Congruence
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2015-10-14 20:35
PolygonTransformationGeometric Construction
This example demonstrates how to prove the congruence of a figure using a sequence of rigid motions. It involves core mathematical concepts such as rigid transformations and geometric congruence. The construction utilizes GeoGebra commands and objects like Polygon, Vector, Translate, Point, Line, Mirror, and Segment. It is suitable for high school mathematics education, helping students understand and apply properties of rigid motions and congruence.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Vector(C, D) → u
  3. Translate(A, u) → A'
  4. Translate(B, u) → B'
  5. Translate(C, u) → C'
  6. Polygon(A', B', C') → poly1', c', a', b'
  7. Point(u) → E
  8. Vector(C, E) → v
  9. Translate(A, v) → A'_1
  10. Translate(B, v) → B'_1
  11. Translate(C, v) → C'_1
  12. Polygon(A'_1, B'_1, C'_1) → poly1'_1, c'_1, a'_1, b'_1
  13. Line(D, F) → d
  14. Mirror(A', d) → A''
  15. Mirror(B', d) → B''
  16. Mirror(C', d) → C''
  17. Polygon(A'', B'', C'') → poly1'', c'', a'', b''
  18. Mirror(A'_1, d) → G
  19. Mirror(B'_1, d) → H
  20. Mirror(C'_1, d) → I
  21. Polygon(G, H, I) → poly2, i, g, h
  22. Mirror(A, d) → A'_2
  23. Mirror(B, d) → B'_2
  24. Mirror(C, d) → C'_2
  25. Polygon(A'_2, B'_2, C'_2) → poly1'_2, c'_2, a'_2, b'_2
  26. Line(J, d) → e
  27. Vector(J, K) → w
  28. Mirror(A'_1, e) → L
  29. Mirror(B'_1, e) → M
  30. Mirror(C'_1, e) → N
  31. Point(w) → O
  32. Vector(J, O) → z
  33. Translate(L, f) → L'
  34. Translate(M, f) → M'
  35. Translate(N, f) → N'
  36. Translate(L', z) → L''
  37. Translate(M', z) → M''
  38. Translate(N', z) → N''
  39. Translate(A'_2, z) → P
  40. Translate(B'_2, z) → Q
  41. Translate(C'_2, z) → R
  42. Mirror(A, e) → A'_3
  43. Mirror(B, e) → B'_3
  44. Mirror(C, e) → C'_3
  45. Polygon(A'_3, B'_3, C'_3) → poly1'_3, c'_3, a'_3, b'_3
  46. Translate(A'_3, z) → S
  47. Translate(B'_3, z) → T
  48. Translate(C'_3, z) → U
  49. Polygon(S, T, U) → poly3, u_1, s, t
  50. Segment(E, D) → j
  51. Mirror(A'_1, d) → V
  52. Mirror(B'_1, d) → W
  53. Mirror(C'_1, d) → Z
  54. Polygon(V, W, Z) → poly4, z_1, v_1, w_1
  55. Vector(A_1, D) → k
  56. Mirror(E, d) → E'
  57. Segment(E', D) → l
  58. Vector(C'_3, D) → m
  59. Segment(A, B) → n
  60. Segment(B, C) → p
  61. Segment(C, A) → q
  62. Translate(C, w) → C'_4
  63. Vector(C, C'_4) → r
  64. Point(r) → B_1
  65. Vector(C, B_1) → a_1
  66. Translate(A, a_1) → A'_4
  67. Translate(B, a_1) → B'_4
  68. Translate(C, a_1) → C'_5
  69. Polygon(A'_4, B'_4, C'_5) → poly1'_4, c'_5, a'_4, b'_4
  70. Segment(B_1, C'_4) → b_1
  71. Translate(A, r) → A'_5
  72. Translate(B, r) → B'_5
  73. Segment(A'_5, B'_5) → n'
  74. Translate(B, r) → B'_6
  75. Translate(C, r) → C'_6
  76. Segment(B'_6, C'_6) → p'
  77. Translate(C, r) → C'_7
  78. Translate(A, r) → A'_6
  79. Segment(C'_7, A'_6) → q'
  80. Mirror(A'_4, e) → C_1
  81. Mirror(B'_4, e) → D_1
  82. Mirror(C'_5, e) → E_1
  83. Polygon(C_1, D_1, E_1) → poly5, e_1, c_1, d_1
  84. Segment(E, D) → f_1
  85. Line(M_1, N_1) → g_1
  86. Segment(O, K) → h_1

AI Drawing Prompt Reference

Draw a polygon and demonstrate its congruence through a sequence of rigid motions like translations and reflections. First, translate the polygon to a new position, then reflect it across a line. Finally, show the congruence through multiple transformations and reflections, highlighting the concepts of rigid motions and congruence.
The prompt above was compiled from the original drawing commands and is for reference only.