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Kite Geometric Construction and Transformation

Kite Geometric Construction and Transformation
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2019-05-25 10:45
CirclePolygonAngleTransformationGeometric Construction
This example demonstrates the dynamic construction of a kite shape using GeoGebra commands, illustrating its geometric properties. Core mathematical concepts include axial symmetry, midpoints, perpendicular bisectors, circles and arcs, rotation and dilation transformations, as well as polygon area and angle relationships. The construction uses commands such as Segment, Midpoint, OrthogonalLine, Point, Ray, Circle, Intersect, CircleSector, Angle, Dilate, Rotate, Polygon, Vector, Translate, and Line to build vertices, edges, auxiliary circles, and angle labels, and applies transformations to generate symmetric or scaled versions. Suitable for middle and high school geometry lessons to explore the definition of a kite, its symmetry, and geometric transformations.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Midpoint(a) → C
  3. OrthogonalLine(C, a) → b
  4. Point(b) → E
  5. Point(b) → G_1
  6. Ray(E, G_1) → h_2
  7. Point(h_2) → D
  8. Segment(C, D) → i
  9. Circle(D, (0.3 * i)) → q
  10. Intersect(q, h_2, 1) → H_1
  11. Segment(A, D) → c
  12. Intersect(q, c, 1) → K
  13. Segment(D, B) → d
  14. Segment(B, E) → e
  15. Segment(E, A) → f
  16. Segment(A, C) → g
  17. Circle(A, (0.3 * g)) → h
  18. Circle(B, (0.3 * g)) → k
  19. Segment(C, E) → j
  20. Circle(E, (0.3 * j)) → p
  21. Circle(C, (0.3 * g)) → r
  22. Intersect(h, f, 1) → F
  23. Intersect(h, c, 1) → G
  24. Intersect(k, e, 1) → H
  25. Intersect(k, d, 1) → I
  26. Intersect(q, d, 1) → J
  27. Intersect(p, e, 1) → L
  28. Intersect(b, j) → M
  29. Intersect(b, j) → N
  30. Intersect(b, j) → O
  31. Intersect(b, j) → P
  32. Intersect(b, j) → Q
  33. Intersect(p, b, 2) → R
  34. Intersect(b, i) → S
  35. Intersect(q, b, 1) → T
  36. Intersect(p, f, 1) → U
  37. CircleSector(B, I, H) → s
  38. Angle(A, D, B) → α
  39. Angle(U, E, L) → β
  40. CircleSector(B, H, I) → t
  41. Dilate(D, If[0 < l ≤ 1, 1 - l, 0], A) → D'
  42. Segment(D, D') → m
  43. Rotate(D, If[1 < l ≤ 2, ((l - 1) * α), α], D) → D'_1
  44. Rotate(D', If[1 < l ≤ 2, ((l - 1) * α), α], D) → D''
  45. Segment(D'_1, D'') → m'
  46. Dilate(A, If[2 < l ≤ 3, 3 - l, 0], E) → A'
  47. Segment(A, A') → n
  48. Rotate(A, If[3 < l ≤ 4, ((-(l - 3)) * ((2 * π) - β)), (-((2 * π) - β))], E) → A'_1
  49. Rotate(A', If[3 < l ≤ 4, ((-(l - 3)) * ((2 * π) - β)), (-((2 * π) - β))], E) → A''
  50. Segment(A'_1, A'') → n'
  51. Polygon(E, B, D, A) → poly1, e_1, b_1, d_1, a_1
  52. CircleSector(E, L, R) → f_1
  53. Dilate(D, If[4 < l ≤ 5, 5 - l, 0], E) → D'_2
  54. Segment(D, D'_2) → g_1
  55. Dilate(B, If[5 < l ≤ 6, 6 - l, 0], A) → B'
  56. Vector(B, B') → u
  57. Translate(s, u) → s'
  58. Line(A, e) → h_1
  59. Intersect(h, h_1, 2) → V
  60. Intersect(h, g, 1) → W
  61. Angle(V, A, G) → γ
  62. Rotate(s', If[6 < l ≤ 7, ((l - 6) * γ), γ], A) → s''
  63. CircleSector(D, K, H_1) → c_1
  64. Rotate(f_1, If[8 < l ≤ 9, ((l - 8) * ((2 * π) - β) / 2), ((2 * π) - β) / 2], E) → p_1
  65. Dilate(A, If[9 < l ≤ 10, 10 - l, 0], C) → A'_2
  66. Segment(A, A'_2) → i_1
  67. Dilate(C, If[10 < l ≤ 11, 11 - l, 0], B) → D'_3
  68. Segment(C, D'_3) → j_1
  69. Segment(B, C) → l_1
  70. Intersect(k, l_1, 1) → Z
  71. CircleSector(B, I, Z) → q_1
  72. CircleSector(A, F, W) → r_1
  73. Rotate(C, If[11.5 < l ≤ 12.5, ((l - 11.5) * π), 3.141592653589793], C) → C'
  74. Rotate(D'_3, If[11.5 < l ≤ 12.5, ((l - 11.5) * π), 3.141592653589793], C) → A_1
  75. Segment(C', A_1) → m_1
  76. Dilate(A, If[12.5 < l ≤ 13.5, 13.5 - l, 0], B) → A'_3
  77. Vector(A, A'_3) → v
  78. Translate(r_1, v) → s_1
  79. Line(B, n) → n_1
  80. Point(n_1) → B_1
  81. Ray(B, B_1) → t_1
  82. Point(t_1) → C_1
  83. Angle(H, B, C_1) → δ
  84. Angle(Z, B, H) → ε
  85. Rotate(s_1, If[13.5 < l ≤ 14.5, ((-(l - 13.5)) * (ε + δ)), (-(ε + δ))], B) → c_2
  86. Dilate(B, If[14.5 < l ≤ 15.5, 15.5 - l, 0], A) → B'_1
  87. Vector(B, B'_1) → w
  88. Translate(q_1, w) → d_2
  89. Translate(I, w) → I'
  90. Angle(G, A, I') → ζ
  91. Angle(D, B, C) → η
  92. Rotate(d_2, If[15.5 < l ≤ 16.5, ((-(l - 15.5)) * (ζ + η)), (-(ζ + η))], A) → e_2
  93. Midpoint(A, B) → D_1
  94. Segment(D, E) → a_2
  95. Segment(A, B) → b_2
  96. Intersect(r, a_2, 2) → E_1
  97. Intersect(r, b_2, 2) → F_1
  98. CircleSector(C, E_1, F_1) → f_2
  99. Rotate(f_2, If[16.5 < l ≤ 17.5, ((-(l - 16.5)) * π / 2), -1.5707963267948966], D_1) → g_2
  100. Rotate(c_1, If[7 < l ≤ 8, ((l - 7) * α / 2), α / 2], D) → k_1