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Construction and Verification of Rotational Symmetry

Construction and Verification of Rotational Symmetry
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2018-09-24 03:02
PolygonAngleTransformationGeometric Construction
This example demonstrates how to construct a geometric figure with rotational symmetry using GeoGebra. Create an initial polygon (poly1) with the Polygon tool, then use the Rotate command to generate a symmetric counterpart (poly2) around a specified center. Verify the rotational invariance using Midpoint, Line, OrthogonalLine, and Intersect tools. Key commands: Polygon, Rotate, Midpoint, Line, OrthogonalLine, Intersect. Suitable for middle and high school mathematics courses covering geometric transformations, helping students understand rotational symmetry, center of rotation, angles, and symmetry properties while developing spatial reasoning skills.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 5) → poly1, a, b, c, d, e, C, D, E
  2. Polygon(F, N, G, M, H, O, J, L) → poly2, f, n, g, m, h, o, j, l
  3. Segment(I, K) → i
  4. Segment(K, P) → k
  5. Segment(P, Q) → p
  6. Rotate(K, 240°, R) → K'
  7. Rotate(P, 240°, R) → P'
  8. Segment(K', P') → k'
  9. Rotate(K', 240°, R) → K''
  10. Rotate(P', 240°, R) → P''
  11. Segment(K'', P'') → k''
  12. Rotate(P, 240°, R) → P'_1
  13. Rotate(Q, 240°, R) → Q'
  14. Segment(P'_1, Q') → p'
  15. Rotate(I, 120°, R) → I'
  16. Rotate(K, 120°, R) → K'_1
  17. Segment(I', K'_1) → i'
  18. Rotate(I, 240°, R) → I'_1
  19. Rotate(K, 240°, R) → K'_2
  20. Segment(I'_1, K'_2) → i'_1
  21. Rotate(P, 120°, R) → P'_2
  22. Rotate(Q, 120°, R) → Q'_1
  23. Segment(P'_2, Q'_1) → p'_1
  24. Segment(I, Q'_1) → q
  25. Segment(Q, I'_1) → r
  26. Segment(Q', I') → s
  27. Polygon(K, P, Q'_1, I, Q, I'_1, K', P', Q', I', K'', P'') → poly3, k_1, p'', q'_1, i_1, p_1, q_1, i'_2, k'_1, p'_2, q', i'_3, k''_1
  28. Polygon(S, T, U, V) → poly4, s_1, t, u, v
  29. Midpoint(N, O) → W
  30. Midpoint(S, T) → Z
  31. Midpoint(S, V) → A_1
  32. Line(A_1, s_1) → a_1
  33. Line(Z, t) → b_1
  34. Intersect(a_1, b_1) → B_1
  35. OrthogonalLine(D, a) → c_1
  36. OrthogonalLine(C, e) → d_1
  37. Intersect(c_1, d_1) → C_1
  38. Rotate(poly1, w°, C_1) → poly1'
  39. Rotate(F, w°, W) → F'
  40. Rotate(N, w°, W) → N'
  41. Rotate(G, w°, W) → G'
  42. Rotate(M, w°, W) → M'
  43. Rotate(H, w°, W) → H'
  44. Rotate(O, w°, W) → O'
  45. Rotate(J, w°, W) → J'
  46. Rotate(L, w°, W) → L'
  47. Polygon(F', N', G', M', H', O', J', L') → poly2', f', n', g', m', h', o', j', l'
  48. Rotate(K, w°, R) → K'_3
  49. Rotate(P, w°, R) → P'_3
  50. Rotate(Q, w°, R) → Q'_2
  51. Rotate(I'_1, w°, R) → D_1
  52. Rotate(K', w°, R) → K''_1
  53. Rotate(P', w°, R) → P''_1
  54. Rotate(Q', w°, R) → Q''
  55. Rotate(I', w°, R) → I''
  56. Rotate(K'', w°, R) → K'''
  57. Rotate(P'', w°, R) → P'''
  58. Rotate(Q'_1, w°, R) → E_1
  59. Rotate(I, w°, R) → I'_2
  60. Polygon(K'_3, P'_3, E_1, I'_2, Q'_2, D_1, K''_1, P''_1, Q'', I'', K''', P''') → poly3', k'_3, p''', e_1, i'_4, p'_3, q'_2, d_2, k''_2, p''_1, q'', i'', k'''
  61. Rotate(S, w°, B_1) → S'
  62. Rotate(T, w°, B_1) → T'
  63. Rotate(U, w°, B_1) → U'
  64. Rotate(V, w°, B_1) → V'
  65. Polygon(S', T', U', V') → poly4', s', t', u', v'
  66. Polygon(F_1, G_1, H_1) → poly5, h_1, f_1, g_1
  67. OrthogonalLine(G_1, g_1) → j_1
  68. Intersect(j_1, g_1) → I_1
  69. Midpoint(I_1, G_1) → J_1
  70. Rotate(F_1, w°, J_1) → K_1
  71. Rotate(G_1, w°, J_1) → L_1
  72. Rotate(H_1, w°, J_1) → M_1
  73. Polygon(K_1, L_1, M_1) → poly5', m_1, k_2, l_1