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Dynamic Illustration of Equal Areas

Dynamic Illustration of Equal Areas
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2017-02-19 23:01
CirclePolygonTransformation
In collections Perimeter & Area
This example demonstrates the concept of equal areas through dynamic transformations, showing that different plane figures can have the same area. Key mathematical concepts include area calculation and geometric transformations. It uses GeoGebra objects such as circles, polygons, and vectors for construction. Suitable for middle school and above students to learn area calculation and geometric transformations.

GeoGebra Commands & Parameters

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Circle(B, c) → e
  4. Point(e) → C
  5. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  8. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  9. Segment(B, B') → n
  10. Segment(B, B'_1) → p
  11. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  12. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  13. Segment(B'_2, L) → p'
  14. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  15. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  16. Segment(B'_3, B'') → n'
  17. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  18. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  19. Segment(M, L') → p''
  20. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  21. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  22. Segment(N, B''') → n''
  23. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  24. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  25. Segment(N', O) → n'''
  26. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  27. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  28. Segment(M', L'') → p'''
  29. Segment(F, G) → q
  30. Segment(F, B) → r
  31. Segment(A, B) → s
  32. Min(r, s) → o
  33. Circle(F, (0.15 * o)) → t
  34. Intersect(l, q) → P
  35. Intersect(t, q) → Q,
  36. Intersect(t, r) → R,
  37. CircleSector(F, Q, R) → c_1
  38. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  39. Vector(F, F') → u
  40. Translate(c_1, u) → d_1
  41. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  42. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  43. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  44. Vector(F, F'_1) → v
  45. Vector(C, C') → w
  46. Translate(e_1, w) → f_1
  47. Translate(c_1, v) → g_1
  48. Line(A, p) → h_1
  49. Line(E, p''') → i_1
  50. Point(h_1) → S
  51. Angle(S, A, E) → α
  52. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  53. Line(E, h_1) → j_1
  54. Point(j_1) → T
  55. Angle(D, E, T) → β
  56. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  57. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  58. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  59. Vector(E, E') → u_1
  60. Vector(F, F'_2) → v_1
  61. Translate(p_1, u_1) → q_1
  62. Translate(c_1, v_1) → r_1
  63. Line(G, n'') → l_1
  64. Point(l_1) → U
  65. Line(G, n') → m_1
  66. Point(m_1) → V
  67. Angle(C, G, V) → γ
  68. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  69. Line(D, p) → n_1
  70. Point(n_1) → W
  71. Angle(W, D, B) → δ
  72. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  73. Midpoint(A, B) → Z
  74. Midpoint(B, C) → A_1
  75. Midpoint(Z, B) → B_1
  76. Midpoint(A_1, B) → C_1
  77. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  78. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  79. Vector(A, A') → w_1
  80. Vector(C, C'_1) → a_1
  81. Translate(e_1, a_1) → c_2
  82. Translate(k_1, w_1) → d_2
  83. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  84. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  85. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  86. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  87. Vector(B_1, E_1) → b_1
  88. Vector(C_1, D_1) → u_2
  89. Translate(e_2, b_1) → g_2
  90. Translate(f_2, u_2) → h_2
  91. Segment(D, C) → a_3
  92. Segment(A, F) → b_3
  93. Line(A, B) → f_4
  94. Polygon(A, F, B) → poly4, b_4, a_4, f_5
  95. Polygon(B, C, D) → poly5, d_4, b_5, c_4
  96. OrthogonalLine(F, f_4) → h_4
  97. Line(D, B) → g_4
  98. OrthogonalLine(C, g_4) → i_4
  99. Intersect(f_4, h_4) → R_1
  100. Intersect(g_4, i_4) → S_1
  101. Dilate(B, If[13 < a ≤ 14, 14 - a, 0], R_1) → B'_4
  102. Dilate(F, If[13 < a ≤ 14, 14 - a, 0], R_1) → F'_3
  103. Angle(B, R_1, F) → ε
  104. Segment(B, B'_4) → i_2
  105. Segment(F, F'_3) → j_2
  106. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], F) → A'_1
  107. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], C) → D'
  108. Segment(D, D') → k_2
  109. Segment(A, A'_1) → l_2
  110. Dilate(C, If[14 < a ≤ 15, 15 - a, 0], S_1) → C'_2
  111. Dilate(D, If[14 < a ≤ 15, 15 - a, 0], S_1) → D'_1
  112. Segment(D, D'_1) → m_2
  113. Segment(C, C'_2) → n_2
  114. Angle(D, S_1, C) → ζ
  115. Dilate(F'_3, If[15 < a ≤ 16, 16 - a, 0], C) → H
  116. Vector(F'_3, H) → v_2
  117. Translate(F, v_2) → F'_4
  118. Translate(F'_3, v_2) → I
  119. Segment(F'_4, I) → p_2
  120. Line(C, j_2) → q_2
  121. Point(q_2) → J
  122. Angle(J, C, C'_2) → η
  123. Rotate(F'_4, If[16 < a ≤ 17, ((a - 16) * η), η], C) → K
  124. Rotate(I, If[16 < a ≤ 17, ((a - 16) * η), η], C) → I'
  125. Segment(K, I') → r_2