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Reflections, Rotations, Dilations, and Translations

Reflections, Rotations, Dilations, and Translations
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2020-02-24 08:25
VectorCoordinate GeometryTransformationGeometric Construction
This example demonstrates the effects of geometric transformations like reflections, rotations, dilations, and translations on shapes in a 2D plane. It involves core math concepts including transformations, coordinate geometry, vectors, and parametric equations. The construction utilizes GeoGebra commands such as Intersect, Point, Circle, Segment, Polygon, Rotate, Vector, Translate, Mirror, Dilate, and Midpoint. Suitable for high school and above students to understand the properties and applications of geometric transformations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(yAxis) → B
  3. Circle(A, B) → c
  4. Segment(A, C) → a
  5. Segment(C, D) → b
  6. Segment(D, E) → d
  7. Segment(E, A) → e
  8. Polygon(A, C, D, E) → poly1, a_1, c_1, d_1, e_1
  9. Rotate(A, Windmill, A) → A'
  10. Rotate(C, Windmill, A) → C'
  11. Rotate(D, Windmill, A) → D'
  12. Rotate(E, Windmill, A) → E'
  13. Polygon(A', C', D', E') → poly1', a', c', d', e'
  14. Polygon(A, F, G, H) → poly2, a_2, f, g, h
  15. Polygon(A, I, J, K) → poly3, a_3, i, j, k
  16. Polygon(A, L, M, N) → poly4, a_4, l, m, n
  17. Rotate(A, Windmill, A) → A'_1
  18. Rotate(F, Windmill, A) → F'
  19. Rotate(G, Windmill, A) → G'
  20. Rotate(H, Windmill, A) → H'
  21. Polygon(A'_1, F', G', H') → poly2', a'_1, f', g', h'
  22. Rotate(A, Windmill, A) → A'_2
  23. Rotate(I, Windmill, A) → I'
  24. Rotate(J, Windmill, A) → J'
  25. Rotate(K, Windmill, A) → K'
  26. Polygon(A'_2, I', J', K') → poly3', a'_2, i', j', k'
  27. Rotate(A, Windmill, A) → A'_3
  28. Rotate(L, Windmill, A) → L'
  29. Rotate(M, Windmill, A) → M'
  30. Rotate(N, Windmill, A) → N'
  31. Polygon(A'_3, L', M', N') → poly4', a'_3, l', m', n'
  32. Polygon(A, O, P, Q, R) → poly5, a_5, o, p, q, r_3
  33. Polygon(S, T, U, V) → poly6, s, t, u, v
  34. Circle(W, Z) → f_1
  35. Polygon(A_1, B_1, K_1, L_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1) → poly7, a_6, j_1, k_1, l_1, b_1, c_2, d_2, e_2, f_2, g_1, h_1, i_1
  36. Polygon(M_1, N_1, W_1, Z_1, A_2, B_2, C_2, O_1, P_1, Q_1, R_1, S_1, T_1, U_1, V_1) → poly8, m_1, v_1, w_1, z_1, a_7, b_2, c_3, n_1, o_1, p_1, q_1, r_1, s_1, t_1, u_1
  37. Point(Circle[A, Clouds]) → E_2
  38. Segment(A, E_2) → g_2
  39. Vector(A, E_2) → z
  40. Translate(A_1, z) → F_2
  41. Translate(B_1, z) → G_2
  42. Translate(C_1, z) → H_2
  43. Translate(D_1, z) → I_2
  44. Translate(E_1, z) → J_2
  45. Translate(F_1, z) → K_2
  46. Translate(G_1, z) → L_2
  47. Translate(H_1, z) → M_2
  48. Translate(I_1, z) → N_2
  49. Translate(J_1, z) → O_2
  50. Translate(K_1, z) → P_2
  51. Translate(L_1, z) → Q_2
  52. Polygon(F_2, G_2, P_2, Q_2, H_2, I_2, J_2, K_2, L_2, M_2, N_2, O_2) → poly7', f_3, o_2, p_2, q_2, g_3, h_2, i_2, j_2, k_2, l_2, m_2, n_2
  53. Translate(M_1, z) → R_2
  54. Translate(N_1, z) → S_2
  55. Translate(O_1, z) → T_2
  56. Translate(P_1, z) → U_2
  57. Translate(Q_1, z) → V_2
  58. Translate(R_1, z) → W_2
  59. Translate(S_1, z) → Z_2
  60. Translate(T_1, z) → A_3
  61. Translate(U_1, z) → B_3
  62. Translate(V_1, z) → C_3
  63. Translate(W_1, z) → D_3
  64. Translate(Z_1, z) → E_3
  65. Translate(A_2, z) → F_3
  66. Translate(B_2, z) → G_3
  67. Translate(C_2, z) → H_3
  68. Polygon(R_2, S_2, D_3, E_3, F_3, G_3, H_3, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, C_3) → poly8', r_2, c_4, d_3, e_3, f_4, g_4, h_3, s_2, t_2, u_2, v_2, w_2, z_2, a_8, b_3
  69. Point(s) → D_2
  70. Point(s) → I_4
  71. Polygon(D_2, G_4, H_4, I_4) → poly12, d_5, g_5, h_4, i_4
  72. Point(h_4) → M_4
  73. Polygon(H_4, J_4, K_4, L_4, M_4) → poly13, h_5, j_4, k_4, l_5, m_4
  74. Mirror(H_4, h_4) → N_4
  75. Mirror(J_4, h_4) → O_4
  76. Mirror(K_4, h_4) → P_4
  77. Mirror(L_4, h_4) → Q_4
  78. Mirror(M_4, h_4) → R_4
  79. Polygon(N_4, O_4, P_4, Q_4, R_4) → poly13', n_4, o_4, p_3, q_4, r_4
  80. Point(s) → T_4
  81. Point(s) → U_4
  82. Polygon(S_4, T_4, U_4, V_4) → poly14, s_4, t_4, u_4, v_4
  83. Polygon(W_4, Z_4, A_5, B_5, C_5, D_5, E_5, F_5, G_5, H_5) → poly15, w_4, h_6, z_4, a_{10}, b_5, c_6, d_6, e_5, f_6, g_6
  84. Point(h_6) → I_5
  85. Dilate(W_4, Balloon, I_5) → J_5
  86. Dilate(Z_4, Balloon, I_5) → K_5
  87. Dilate(A_5, Balloon, I_5) → L_5
  88. Dilate(B_5, Balloon, I_5) → M_5
  89. Dilate(C_5, Balloon, I_5) → N_5
  90. Dilate(D_5, Balloon, I_5) → O_5
  91. Dilate(E_5, Balloon, I_5) → P_5
  92. Dilate(F_5, Balloon, I_5) → Q_5
  93. Dilate(G_5, Balloon, I_5) → R_5
  94. Dilate(H_5, Balloon, I_5) → S_5
  95. Polygon(J_5, K_5, L_5, M_5, N_5, O_5, P_5, Q_5, R_5, S_5) → poly15', j_5, s_5, k_5, l_6, m_5, n_5, o_5, p_5, q_5, r_5
  96. Segment(S_4, J_5) → b_6
  97. Segment(V_4, S_5) → e_6
  98. Point(n_5) → V_5
  99. Segment(L_5, Q_5) → j_6
  100. Midpoint(v_4) → I_3
  101. Point(v_4) → J_3
  102. Point(v_4) → K_3
  103. Segment(L_5, J_3) → i_3
  104. Segment(I_3, V_5) → j_3
  105. Segment(K_3, Q_5) → k_3

AI Drawing Prompt Reference

Draw a plane figure with a circle and a windmill shape. First, draw a circle and select four points A, C, D, E on it, then connect them to form a quadrilateral. Next, rotate this quadrilateral around point A to generate four new quadrilaterals. Then, draw a windmill shape and rotate each vertex of the windmill around point A to generate new windmill shapes. Finally, draw a line segment from point A to a point E2 on the circle, generate a vector, and use this vector to translate each vertex of the windmill to generate new windmill shapes.
The prompt above was compiled from the original drawing commands and is for reference only.