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Kite Geometric Construction: From Segments to Composite Transformations

Kite Geometric Construction: From Segments to Composite Transformations
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2020-01-14 14:26
PolygonAngleVectorTransformationGeometric Construction
This GeoGebra example demonstrates a complex kite-shaped geometric construction built using a variety of commands including Segment, Midpoint, OrthogonalLine, Point, Ray, Circle, Intersect, CircleSector, Angle, Dilate, Rotate, Polygon, Vector, Translate, and Line. The core mathematical concepts involve the definition of a kite (two pairs of adjacent equal sides), symmetry, perpendicular diagonals, angle relationships, similarity and congruence transformations (dilation, rotation, translation), and vector applications. The construction begins with segment AB and its midpoint to create perpendicular lines, then uses circles and rays to determine key points such as C, D, and E. CircleSector generates sectors, Angle marks angle α, Dilate and Rotate apply scaling and rotation, Polygon builds polygons, and Vector with Translate performs translations, resulting in a composite figure containing triangles, trapezoids, and rhombi. This resource is suitable for middle and high school geometry classes to explore geometric transformations, properties of shapes, and dynamic construction techniques, and also serves as an advanced GeoGebra practice for students and teachers.

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