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Interactive Discovery of Trapezoid Area

Interactive Discovery of Trapezoid Area
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2021-11-05 09:13
PolygonGeometric Construction
In collections Visual Geometry for Grade 9
This example demonstrates the derivation of the trapezoid area formula through interactive exploration. Key math concepts include trapezoid definition, area calculation, vectors, and geometric transformations. GeoGebra commands like Ray, Point, Segment, Midpoint, Rotate, Translate, and objects such as vectors and polygons are used in the construction, suitable for high school students to understand geometric properties and formula derivation.

GeoGebra Commands & Parameters

  1. Ray(A, B) → g
  2. Point(g) → C
  3. Segment(A, C) → i
  4. Slope(i) → m
  5. PointIn(a) → D
  6. Midpoint(D, A) → S
  7. Line(A, B) → f
  8. Ray(A, C) → h
  9. Line(D, i) → j
  10. Point(j) → E
  11. Ray(D, E) → k
  12. Point(k) → F
  13. Segment(D, F) → l
  14. Segment(A, D) → n
  15. Segment(F, C) → p
  16. Midpoint(p) → G
  17. Dilate(A, If[0 < c ≤ 1, 1 - c, 0], C) → A'
  18. Segment(A, A') → q
  19. Dilate(C, If[1 < c ≤ 2, 2 - c, 0], G) → C'
  20. Dilate(G, If[2 < c ≤ 3, 3 - c, 0], F) → G'
  21. Segment(G, G') → s
  22. Rotate(C, If[3 < c ≤ 4, ((c - 3) * π), π], G) → C'_1
  23. Rotate(C', If[3 < c ≤ 4, ((c - 3) * π), π], G) → C''
  24. Dilate(F, If[4 < c ≤ 5, 5 - c, 0], D) → F'
  25. Segment(F, F') → t
  26. Dilate(D, If[5 < c ≤ 6, 6 - c, 0], S) → D'
  27. Dilate(D, If[6.5 < c ≤ 7.5, 7.5 - c, 0], G) → D'_1
  28. Polygon(A, C, G, S) → poly1, e, f_1, g_1, k_1
  29. Rotate(G, If[8 < c ≤ 9, ((-(c - 8)) * π), π], G) → G'_1
  30. Rotate(F, If[8 < c ≤ 9, ((-(c - 8)) * π), π], G) → F'_1
  31. Rotate(D, If[8 < c ≤ 9, ((-(c - 8)) * π), π], G) → D'_2
  32. Rotate(D, If[8 < c ≤ 9, ((-(c - 8)) * π), π], G) → D'_3
  33. Rotate(D'_1, If[8 < c ≤ 9, ((-(c - 8)) * π), π], G) → H
  34. Rotate(C'_1, If[8 < c ≤ 9, ((-(c - 8)) * π), π], G) → I
  35. Rotate(C'', If[8 < c ≤ 9, ((-(c - 8)) * π), π], G) → C'''
  36. OrthogonalLine(D, q) → i_1
  37. Intersect(f, q) → J
  38. Dilate(D, If[6 < c ≤ 7, 7 - c, 0], J) → D'_4
  39. Segment(D, D'_4) → j_1
  40. Min(t, q) → o
  41. Circle(J, (0.15 * o)) → e_1
  42. Intersect(e_1, i_1) → K, L
  43. Intersect(e_1, f) → M, N
  44. Angle(D, J, A) → α
  45. Angle(C, A, D) → β
  46. Intersect(i_1, f) → O
  47. Dilate(D, If[6 < c ≤ 7, 7 - c, 0], O) → D'_5
  48. Point(f) → P
  49. Point(f) → Q
  50. Circle(C, t) → p_1
  51. Intersect(p_1, f, 2) → R
  52. Ray(R, Q) → m_1
  53. Angle(F, C, A) → δ
  54. Line(A, D) → l_1
  55. Rotate(D, If[6 < c ≤ 7, ((-(c - 6)) * π), (-π)], S) → D'_6
  56. Rotate(D', If[6 < c ≤ 7, ((-(c - 6)) * π), (-π)], S) → D''
  57. Segment(F, C) → r
  58. Segment(D, A) → d
  59. Midpoint(D, O) → T
  60. Polygon(S, G, F, D) → q1, s_1, g_2, f_2, d_1
  61. Circle(A, (0.4 * Distance[A, S])) → c_1
  62. Intersect(c_1, d, 1) → U
  63. Intersect(c_1, q, 1) → V
  64. Dilate(D, If[5 < c ≤ 6, 6 - c, 0], T) → D'_7
  65. Segment(D, D'_7) → n_1
  66. Dilate(T, If[6 < c ≤ 7, 7 - c, 0], O) → T'
  67. Segment(T, T') → q_1
  68. Angle(C, O, D) → γ
  69. Rotate(D, If[7 < c ≤ 8 - ((c - 7) * π), (-π)], T) → D'_8
  70. Rotate(D'_7, If[7 < c ≤ 8 - ((c - 7) * π), (-π)], T) → W
  71. Rotate(D, If[7 < c ≤ 8, ((-(c - 7)) * π), (-π)], T) → D'_9
  72. Rotate(D'_7, If[7 < c ≤ 8, ((-(c - 7)) * π), (-π)], T) → Z
  73. Segment(D'_9, Z) → r_1
  74. Dilate(T, If[8 < c ≤ 9, 9 - c, 0], S) → T'_1
  75. Dilate(T, If[8 < c ≤ 9, 9 - c, 0], G) → T'_2
  76. Segment(T, T'_2) → t_1
  77. Segment(T, T'_1) → a_1
  78. Angle(V, A, U) → ε
  79. Rotate(V, If[9 < c ≤ 10, ((c - 9) * ε), ε], A) → V'
  80. CircleSector(A, V, V') → h_1
  81. Dilate(A, If[10.5 < c ≤ 11.5, 11.5 - c, 0], S) → A'_1
  82. Vector(A, A'_1) → u
  83. Translate(h_1, u) → c_2
  84. Rotate(S, If[12 < c ≤ 13, ((-(c - 12)) * π), π], G) → S'
  85. Rotate(G, If[12 < c ≤ 13, ((-(c - 12)) * π), π], G) → G'_2
  86. Rotate(F, If[12 < c ≤ 13, ((-(c - 12)) * π), π], G) → F'_2
  87. Rotate(D, If[12 < c ≤ 13, ((-(c - 12)) * π), π], G) → D'_{10}
  88. Polygon(S', G'_2, F'_2, D'_{10}) → q1', s', g'_2, f'_2, d'_{10}
  89. Rotate(F, If[12 < c ≤ 13, ((-(c - 12)) * π), π], G) → F'_3
  90. Rotate(F', If[12 < c ≤ 13, ((-(c - 12)) * π), π], G) → F''
  91. Segment(F'_3, F'') → t'
  92. Segment(G, F'_2) → b_1
  93. Rotate(S, If[12 < c ≤ 13, ((-(c - 12)) * π), π], G) → S'_1
  94. Segment(S'_1, D'_{10}) → h_2
  95. Segment(S'_1, G) → i_2
  96. Segment(F, G) → j_2
  97. Segment(G, C) → k_2
  98. Dilate(A, If[13 < c ≤ 14, 14 - c, 0], D'_{10}) → A'_2
  99. Vector(A, A'_2) → v
  100. Translate(h_1, v) → d_2

AI Drawing Prompt Reference

Draw a trapezoid by first drawing a ray g from point A through point B, then taking point C on g and connecting AC to form segment i, calculating the slope m of i. Take point D on a, connect the midpoint S of DA, draw line AB as f, draw ray C from A as h, draw line i from D as j, take point E on j, draw ray k from E, draw segment l from EF, draw segment n from AD, draw segment p from FC, find the midpoint G of p, dilate A along If to A', draw segment AA' as q, dilate C along If to C', dilate G along If to G', draw segment GG' as s, rotate C around G to C'_1, rotate C' around G to C'', rotate F around G to F', draw segment FF' as t, dilate D along If to D', dilate D along If to D'_1, rotate D'_1 around G to H, rotate C'_1 around G to I, rotate C'' around G to C''', draw a perpendicular line i_1 from D to q, intersect f and q at J, dilate D along If to D'_4, draw segment DD'_4 as j_1, find the smaller value o of t and q, draw a circle e_1 with J as center and 0.15 times o as radius, intersect i_1 and e_1 at K and L, intersect e_1 and f at M and N, calculate ∠DJA as α, calculate ∠CAD as β, draw a perpendicular line i_1 from D to q, intersect f and i_1 at O, dilate D along If to D'_5, take point P on f.
The prompt above was compiled from the original drawing commands and is for reference only.