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Trapezium Area Demonstration: Rotation and Definite Integral Visualization

Trapezium Area Demonstration: Rotation and Definite Integral Visualization
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2020-01-16 08:33
PolygonFunctionIntegralTransformationGeometric Construction
① This example dynamically demonstrates the derivation of the trapezium area formula through rotation and reassembly, linking it to the concept of definite integrals. ② Core mathematical concepts include trapezium area, polygon rotation, midpoints, definite integrals, function graphs, and area invariance under transformation. ③ Construction highlights: using GeoGebra commands such as OrthogonalLine, Point, Line, Segment, Midpoint, Polygon, Rotate, Intersect, and If, the construction creates a trapezium and its rotated copy, then rotates the copy to form a parallelogram or rectangle, visually proving the area formula; key objects include the rotated polygon poly1', auxiliary points F, G, H, I, J, K, L, and segments i, j, k, l, m. ④ Suitable for middle school geometry, introductory calculus demonstrations, and interactive teacher resources to help students grasp the connection between area formulas and integration.

GeoGebra Commands & Parameters

  1. OrthogonalLine(A, a) → b
  2. Point(b) → B
  3. Line(C, b) → c
  4. Point(c) → D
  5. Segment(C, A) → d
  6. Segment(B, D) → f
  7. Segment(A, B) → g
  8. Segment(C, D) → h
  9. Midpoint(f) → E
  10. Polygon(A, B, D, C) → poly1, a_1, b_1, d_1, c_1
  11. Rotate(A, Rotate, E) → A'
  12. Rotate(B, Rotate, E) → B'
  13. Rotate(D, Rotate, E) → D'
  14. Rotate(C, Rotate, E) → C'
  15. Polygon(A', B', D', C') → poly1', a', b', d', c'
  16. Midpoint(C, A) → F
  17. Midpoint(B, D) → G
  18. OrthogonalLine(F, h) → j
  19. OrthogonalLine(G, h) → k
  20. Intersect(k, h) → H
  21. Intersect(j, h) → I
  22. Intersect(k, g) → J
  23. Intersect(j, g) → K
  24. Line(C, D) → l
  25. Line(A, B) → m
  26. Intersect(j, l) → L
  27. Intersect(k, m) → M
  28. Intersect(j, m) → N
  29. Intersect(k, l) → O
  30. Point(k) → P
  31. Polygon(P, O, D) → poly2, d_2, p, o
  32. Point(j) → Q
  33. Polygon(N, Q, A) → poly3, a_2, n, q
  34. Polygon(B, P, M) → poly4, m_1, b_2, p_1
  35. Polygon(Q, L, C) → poly5, c_2, q_1, l_1
  36. Rotate(P, α, G) → P'
  37. Rotate(O, α, G) → O'
  38. Rotate(D, α, G) → D'_1
  39. Polygon(P', O', D'_1) → poly2', d'_1, p', o'
  40. Rotate(N, α, F) → N'
  41. Rotate(Q, α, F) → Q'
  42. Rotate(A, α, F) → A'_1
  43. Polygon(N', Q', A'_1) → poly3', a'_1, n', q'
  44. Rotate(B, (-α), G) → B'_1
  45. Rotate(P, (-α), G) → P'_1
  46. Rotate(M, (-α), G) → M'
  47. Polygon(B'_1, P'_1, M') → poly4', m', b'_1, p'_1
  48. Rotate(Q, (-α), F) → Q'_1
  49. Rotate(L, (-α), F) → L'
  50. Rotate(C, (-α), F) → C'_1
  51. Polygon(Q'_1, L', C'_1) → poly5', c'_1, q'_1, l'
  52. If(x(A) > x(C), If[x(B) < x(D), Polygon[A, B, G, O, L, F], Polygon[A, M, G, D, L, F]], If[x(B) < x(D), Polygon[N, B, G, O, C, F], Polygon[N, M, G, D, C, F]]) → polyHex
  53. If(x(A) > x(C), If[x(B) < x(D), Polygon[A, B, G, O, L, F], Polygon[A, M, G, D, L, F]], If[x(B) < x(D), Polygon[N, B, G, O, C, F], Polygon[N, M, G, D, C, F]]) → polyHexBorder
  54. Midpoint(B, D) → R
  55. Line(A, R) → s
  56. Line(C, D) → t
  57. Intersect(t, s) → S
  58. Polygon(A, B, R) → poly6, r_1, a_3, b_3
  59. Rotate(A, (-β), R) → A'_2
  60. Rotate(B, (-β), R) → B'_2
  61. Rotate(R, (-β), R) → R'
  62. Polygon(A'_2, B'_2, R') → poly6', r', a'_2, b'_2
  63. Polygon(A, C, D, R) → poly7, a_4, c_3, d_3, r_2
  64. Polygon(A, R, D, C) → poly8, a_5, r_3, d_4, c_4
  65. Polygon(A'_2, B'_2, R) → poly9, r_4, a'_3, b'_3
  66. Segment(F, G) → e_1
  67. Polygon(F, G, B, A) → poly10, f_1, g_1, b_4, a_6
  68. Rotate(F, (-γ), G) → F'
  69. Rotate(G, (-γ), G) → G'
  70. Rotate(B, (-γ), G) → B'_3
  71. Rotate(A, (-γ), G) → A'_3
  72. Polygon(F', G', B'_3, A'_3) → poly10', f', g', b'_4, a'_4
  73. Polygon(C, F, G, D) → poly11, c_5, f_2, g_2, d_5
  74. Midpoint(Bt, Dt) → T
  75. Polygon(A, D, B) → poly12, b_5, a_7, d_6
  76. Rotate(A, (-δ), T) → A'_4
  77. Rotate(D, (-δ), T) → D'_2
  78. Rotate(B, (-δ), T) → B'_4
  79. Polygon(A'_4, D'_2, B'_4) → poly12', b'_5, a'_5, d'_2
  80. Segment(A, D) → h_1
  81. Segment(A, D) → i_1
  82. Polygon(A, C, D) → poly13, d_7, a_8, c_6
  83. Segment(Ah, Dh) → j_1
  84. OrthogonalLine(Ah, c) → k_1
  85. OrthogonalLine(Dh, c) → n_1
  86. Point(k_1) → V
  87. Point(n_1) → W
  88. OrthogonalLine(V, c) → t_1
  89. OrthogonalLine(W, c) → e_2
  90. Intersect(t_1, c) → Z
  91. Intersect(e_2, c) → A_1
  92. Segment(A_1, Z) → s_1

AI Drawing Prompt Reference

Draw a trapezoid ABCD, connect diagonals AC and BD, and find the midpoint E. Rotate the trapezoid ABCD 180 degrees around point E to get trapezoid A'B'C'D'. Draw perpendicular lines from midpoints F and G to line segments CD and AB, respectively, and let them intersect at points H and I. Connect points H and I, and let them intersect line segment CD at point J and line segment AB at point K. Draw auxiliary line segments l and m, and let them intersect at points L and M. Rotate the trapezoid A'B'C'D' 180 degrees around point E to get trapezoid A''B''C''D''.
The prompt above was compiled from the original drawing commands and is for reference only.