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Geometric Transformations in a Parallelogram

Geometric Transformations in a Parallelogram
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2017-09-13 18:50
PolygonTransformationGeometric Construction
This example illustrates a problem involving geometric transformations within a parallelogram. Key mathematical concepts include rotation, translation, and dilation. Using GeoGebra commands such as Rotate, Translate, and Dilate, dynamic demonstrations of these transformations are provided. Suitable for students at the junior high school level and above.

GeoGebra Commands & Parameters

  1. Rotate(B, a°, A) → B'
  2. Angle(B, A, B') → α
  3. Ray(A, B) → f
  4. Ray(A, B') → g
  5. Line(B, g) → h
  6. Point(g) → C
  7. Line(C, f) → i
  8. Intersect(h, i) → D
  9. Segment(C, D) → j
  10. Segment(A, B) → k
  11. Segment(B, D) → l
  12. Segment(A, C) → m
  13. Min(m, k) → b
  14. Circle(C, (0.2 * b)) → d
  15. Circle(A, (0.2 * b)) → e
  16. Circle(B, (0.2 * b)) → p
  17. Circle(D, (0.2 * b)) → c
  18. Intersect(e, k, 1) → E
  19. Intersect(e, m, 1) → F
  20. Intersect(p, k, 1) → G
  21. Intersect(p, l, 1) → H
  22. Intersect(d, g, 2) → I
  23. Ray(C, I) → q
  24. Intersect(c, h, 2) → J
  25. Ray(D, J) → r
  26. Intersect(c, i, 2) → K
  27. Ray(D, K) → s
  28. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  29. Segment(A, A') → t
  30. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
  31. CircleSector(A, E, E') → c_1
  32. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], C) → A'_1
  33. Vector(A, A'_1) → u
  34. Translate(c_1, u) → d_1
  35. Segment(A, A'_1) → f_1
  36. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], D) → C'
  37. Vector(C, C') → v
  38. Translate(d_1, v) → e_1
  39. Segment(C, C') → g_1
  40. Dilate(D, If[4 < o ≤ 5, 5 - o, 0], B) → D'
  41. Segment(D, D') → h_1
  42. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  43. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  44. Vector(A, A'_2) → w
  45. Translate(A, w) → A'_3
  46. Translate(A', w) → A''
  47. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  48. Vector(A, A'_4) → u_1
  49. Translate(A, u_1) → A'_5
  50. Translate(A'_1, u_1) → L
  51. Intersect(d, i) → M, N
  52. Intersect(d, g) → O, P
  53. Angle(A, C, N) → β
  54. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  55. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  56. Vector(C, C'_1) → v_1
  57. Ray(A, B) → q_1
  58. LineBisector(B, A) → r_1
  59. Distance(C, B) → distanceCB
  60. Segment(C, D) → i_1
  61. Point(i_1) → Q
  62. Line(A, D) → j_1
  63. Line(Q, j_1) → k_1
  64. Line(B, Q) → l_1
  65. Line(C, A) → m_1
  66. Intersect(m_1, k_1) → R
  67. Segment(Q, R) → n_1
  68. Intersect(e, j_1) → S, T
  69. Intersect(e, m_1) → U, V
  70. Line(B, R) → p_1
  71. Intersect(j_1, l_1) → W
  72. Intersect(j_1, p_1) → Z
  73. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], Z) → A'_6
  74. Segment(A, A'_6) → s_1
  75. Dilate(Z, If[7 < o ≤ 8, 8 - o, 0], W) → Z'
  76. Segment(Z, Z') → t_1
  77. Dilate(B, If[7 < o ≤ 8, 8 - o, 0], W) → B'_1
  78. Segment(B, B'_1) → f_2
  79. Dilate(W, If[8 < o ≤ 9, 9 - o, 0], D) → W'
  80. Dilate(W, If[8 < o ≤ 9, 9 - o, 0], Q) → W'_1
  81. Segment(W, W') → g_2
  82. Segment(W, W'_1) → h_2
  83. Dilate(Q, If[9 < o ≤ 10, 10 - o, 0], R) → Q'
  84. Segment(Q, Q') → i_2
  85. Angle(D, A, C) → γ
  86. Rotate(T, If[10 < o ≤ 11, ((o - 10) * γ), γ], A) → T'
  87. CircleSector(A, T, T') → e_2
  88. Dilate(A, If[11 < o ≤ 12, 12 - o, 0], R) → A'_7
  89. Vector(A, A'_7) → w_1
  90. Translate(e_2, w_1) → k_2
  91. Dilate(B, If[12 < o ≤ 13, 13 - o, 0], Z) → B'_2
  92. Segment(B, B'_2) → j_2
  93. Dilate(Z, If[13 < o ≤ 14, 14 - o, 0], R) → Z'_1
  94. Segment(Z, Z'_1) → l_2
  95. Midpoint(A, D) → A_1
  96. Rotate(W, If[15 < o ≤ 16, ((o - 15) * π), π], A_1) → W'_2
  97. Rotate(W', If[15 < o ≤ 16, ((o - 15) * π), π], A_1) → W''
  98. Segment(W'_2, W'') → m_2