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Area of Trapezium Demonstration

Area of Trapezium Demonstration
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2020-11-28 00:52
PolygonTransformationGeometric ConstructionSlider Animation
① This example demonstrates the derivation of the trapezium area formula (half the sum of bases times height) through dynamic rotation and rearrangement. ② The core mathematical concept is the area formula for a trapezium, involving properties of polygons, rotation, midpoints, and parallel lines. ③ Construction highlights: using GeoGebra commands such as OrthogonalLine, Point, Line, Segment, Midpoint, Polygon, Rotate, Intersect, and If, the trapezium and its rotated copy are constructed; rotating one part transforms the trapezium into a parallelogram or rectangle, visually proving the area formula. ④ Suitable for middle school geometry lessons, serving as an interactive visual aid for teachers and a self-exploration tool for students to understand the relationship between bases, height, and area.

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

Create an interactive diagram demonstrating the area relation for a trapezium. Start by placing four points and connecting them in order to form the trapezium polygon. Construct key auxiliary elements using perpendicular lines, midpoints on the relevant segments, and intersection points of those lines to create several triangular/polygon regions that will be rearranged. Then apply a dynamic rotation with the same variable angle to the trapezium’s vertices around the specified rotation centers, producing a rotated copy polygon from the rotated points. Ensure the original and rotated shapes are shown together so the area-by-rearrangement idea is visually supported.
The prompt above was compiled from the original drawing commands and is for reference only.