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Congruence Demonstration

Congruence Demonstration
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2020-01-29 04:09
CircleVectorTransformationGeometric Construction
This example demonstrates the construction and properties of congruent figures, showcasing symmetry and invariance through transformations like rotation and reflection. Core mathematical concepts include congruence, rotation, reflection, angles, and vectors. Construction highlights commands and objects such as Circle, Point, Segment, Rotate, Angle, Ray, Intersect, Min, Polygon, CircleSector, Mirror, Line, Dilate, Vector, and Translate. Suitable for high school mathematics, exploring geometric transformations, properties, and introductory applications in coordinate geometry and probability & statistics.

GeoGebra Commands & Parameters

  1. Circle(A, a) → c
  2. Point(c) → B
  3. Segment(A, B) → f
  4. Circle(A, b) → d
  5. Rotate(B, α, A) → B'
  6. Angle(B, A, B') → β
  7. Ray(A, B') → g
  8. Intersect(d, g, 1) → C
  9. Segment(B, C) → h
  10. Segment(A, C) → i
  11. Min(f, i) → e
  12. Circle(A, (0.25 * e)) → k
  13. Intersect(k, f) → D,
  14. Intersect(k, i) → E,
  15. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  16. CircleSector(A, D, E) → p
  17. Mirror(A, h) → A'
  18. Mirror(B, h) → B'_1
  19. Mirror(C, h) → C'
  20. Polygon(A', B'_1, C') → poly1', c', a', b'_1
  21. Mirror(p, h) → p'
  22. Mirror(A, h) → A'_1
  23. Mirror(B, h) → B'_2
  24. Segment(A'_1, B'_2) → f'
  25. Mirror(A, h) → A'_2
  26. Mirror(C, h) → C'_1
  27. Segment(A'_2, C'_1) → i'
  28. Line(C, B) → j
  29. Point(j) → G
  30. Point(j) → H
  31. Ray(B, H) → l
  32. Ray(C, G) → m
  33. Circle(F, a) → q
  34. Point(q) → I
  35. Segment(F, I) → n
  36. Rotate(I, α, F) → I'
  37. Angle(I, F, I') → γ
  38. Circle(F, b) → r
  39. Ray(F, I') → s
  40. Intersect(r, s, 1) → J
  41. Segment(I, J) → t
  42. Segment(F, J) → f_1
  43. Dilate(B, If[0 < u ≤ 1, 1 - u, 0], I) → B'_3
  44. Vector(B, B'_3) → v
  45. Translate(A, v) → A'_3
  46. Translate(B, v) → B'_4
  47. Segment(A'_3, B'_4) → f'_1
  48. Line(B, n) → g_1
  49. Point(g_1) → K
  50. Angle(K, B, A) → δ
  51. Rotate(A'_3, If[1 < u ≤ 2, ((-(u - 1)) * δ), (-δ)], I) → L
  52. Rotate(B'_4, If[1 < u ≤ 2, ((-(u - 1)) * δ), (-δ)], I) → M
  53. Segment(L, M) → h_1
  54. Dilate(A, If[2 < u ≤ 3, 3 - u, 0], F) → A'_4
  55. Vector(A, A'_4) → w
  56. Translate(A, w) → A'_5
  57. Translate(C, w) → C'_2
  58. Segment(A'_5, C'_2) → i'_1
  59. Line(A, f_1) → i_1
  60. Point(i_1) → N
  61. Angle(N, A, C) → ε
  62. Rotate(A'_5, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → O
  63. Rotate(C'_2, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → P
  64. Segment(O, P) → j_1
  65. Translate(p, w) → p'_1
  66. Rotate(p'_1, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → d_1
  67. Dilate(B, If[4 < u ≤ 5, 5 - u, 0], I) → B'_5
  68. Vector(B, B'_5) → u_1
  69. Translate(A, u_1) → A'_6
  70. Translate(B, u_1) → B'_6
  71. Translate(C, u_1) → C'_3
  72. Polygon(A'_6, B'_6, C'_3) → poly1'_1, c'_3, a'_6, b'_6
  73. Line(B, t) → k_1
  74. Point(k_1) → Q
  75. Angle(Q, B, C) → ζ
  76. Rotate(A'_6, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → R
  77. Rotate(B'_6, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → S
  78. Rotate(C'_3, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → T
  79. Polygon(R, S, T) → poly2, t_1, r_1, s_1
  80. Segment(B, A) → l_1
  81. Segment(A, C) → m_1
  82. Segment(C, B) → n_1
  83. CircleSector(A, D, E) → e_1