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Trapezoid Area Dynamic Demonstration

Trapezoid Area Dynamic Demonstration
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2020-08-14 08:11
PolygonTransformationGeometric Construction
This GeoGebra example dynamically demonstrates the geometric derivation and intuitive meaning of the trapezoid area formula. It involves core mathematical concepts such as the trapezoid area formula (half the sum of the parallel sides multiplied by the height), geometric transformation, symmetry, and shape decomposition. Construction utilizes key GeoGebra commands including OrthogonalLine, Point, Line, Segment, Midpoint, Polygon, Rotate, Intersect, and If, manipulating vertices, sides, midpoints, and rotated copies to build a dynamic trapezoid whose area remains invariant or can be recomposed for proof. Suitable for middle and high school geometry classes, it serves as a visual aid for area formula derivation, transformation exploration, and dynamic geometry practice.

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 with AD and BC being parallel. First, construct a line b perpendicular to line a through point A, determining point B. Then, construct a line c parallel to b through point C, determining point D. Connect points A and B, C and D, and B and D to form trapezoid ABCD. Next, find the midpoint E of AD and rotate trapezoid ABCD around point E to form trapezoid A'B'C'D'. Then, find the midpoints F of AC and G of BD, construct a line j perpendicular to h through F, and construct a line k perpendicular to h through G, determining points H and I. Connect points C and D, A and B, and B and D, find the intersection of line j and h at point L, the intersection of line k and h at point M, the intersection of line j and g at point N, and the intersection of line k and m at point O, determining points P and Q. Finally, construct four new trapezoids through points P, O, and D, Q, L, and A, B, P, and M, Q, N, and C, P, O, and D', Q, L, and A', B', P', and M, Q, N', and C'.
The prompt above was compiled from the original drawing commands and is for reference only.