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Congruent Figures: Dynamic Illustration

Congruent Figures: Dynamic Illustration
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2017-10-12 03:05
TriangleAngleVectorTransformationGeometric Construction
This GeoGebra example offers a dynamic and modifiable illustration of congruent figures, focusing on triangles to demonstrate the definition of congruence through transformations like translation and rotation. The construction utilizes commands such as Segment, Circle, Point, Angle, Rotate, Ray, Intersect, Line, Dilate, Vector, Polygon, Translate, If, and Mirror, creating objects that enable interactive manipulation. Key mathematical concepts include congruence, geometric transformations, angle equality, and corresponding sides. It is designed for middle school or high school geometry education, providing a hands-on way to explore congruence properties dynamically.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, A) → c
  4. Circle(D, (k * a)) → d
  5. Point(d) → E
  6. Segment(D, E) → e
  7. Circle(E, (k * b)) → f
  8. Angle(A, B, C) → α
  9. Rotate(D, α, E) → D'
  10. Angle(D, E, D') → β
  11. Ray(E, D') → g
  12. Intersect(f, g, 1) → F
  13. Angle(C, A, B) → γ
  14. Rotate(E, (-γ), D) → E'
  15. Angle(E', D, E) → δ
  16. Ray(D, E') → h
  17. Segment(D, E') → i
  18. Segment(E', E) → j
  19. Line(E, D') → l
  20. Line(E, D') → n
  21. Circle(D, (k * c)) → p
  22. Ray(D, E') → m
  23. Intersect(p, m, 1) → G
  24. Segment(E, G) → q
  25. Segment(D, G) → r
  26. Dilate(D, If[0 < s ≤ 1, 1 - s], A) → D'_1
  27. Vector(D, D'_1) → u
  28. Polygon(A, C, B) → poly1, b_1, a_1, c_1
  29. Polygon(D, E, G) → poly2, g_1, d_1, e_1
  30. Translate(D, u) → D'_2
  31. Translate(E, u) → E'_1
  32. Translate(G, u) → G'
  33. Polygon(D'_2, E'_1, G') → poly2', g', d'_2, e'_1
  34. Vector(D, A) → v
  35. Translate(D, v) → D'_3
  36. Translate(E, v) → E'_2
  37. Translate(G, v) → G'_1
  38. Polygon(D'_3, E'_2, G'_1) → poly2'_1, g'_1, d'_3, e'_2
  39. Segment(D'_1, G') → t
  40. Segment(E'_1, D'_1) → h_1
  41. Segment(A, G'_1) → i_1
  42. Segment(G'_1, E'_2) → j_1
  43. Segment(E'_2, A) → k_1
  44. Angle(C, A, E'_2) → ε
  45. If(ε > γ, ε - γ, γ - ε) → ζ
  46. Rotate(D'_3, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → H
  47. Rotate(E'_2, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → I
  48. Rotate(G'_1, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → J
  49. Polygon(H, I, J) → poly3, j_2, h_2, i_2
  50. Segment(A, J) → l_1
  51. Segment(A, I) → m_1
  52. Segment(I, J) → n_1
  53. Dilate(J, If[2 < s ≤ 3, 3 - s, 0], C) → J'
  54. Dilate(I, If[2 < s ≤ 3, 3 - s, 0], B) → I'
  55. Segment(I', J') → p_1
  56. Angle(A, I, J) → θ
  57. Angle(D'_2, E'_1, G') → ι
  58. Angle(J, A, I) → λ
  59. Angle(G', D'_1, E'_1) → μ
  60. Polygon(A, J', I') → poly4, i', a_2, j'
  61. Segment(G'_1, E'_2) → q_1
  62. Segment(G', E'_1) → f_1
  63. Rotate(D'_3, ((s - 1) * ζ), A) → K
  64. Rotate(D'_3, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → L
  65. Rotate(E'_2, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → M
  66. Rotate(G'_1, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → N
  67. Polygon(L, M, N) → poly5, n_2, l_2, m_2
  68. Dilate(M, If[2 < s ≤ 3, 3 - s, 0], B) → M'
  69. Dilate(N, If[2 < s ≤ 3, 3 - s, 0], C) → N'
  70. Segment(M', N') → r_1
  71. Polygon(A, M', N') → poly6, n', a_3, m'
  72. Angle(A, M', N') → ο
  73. Angle(A, M, N) → ρ
  74. Segment(A, M) → t_1
  75. Segment(M, N) → b_2
  76. Segment(A, N) → c_2
  77. Angle(B, A, C) → σ
  78. Angle(G, E, D) → τ
  79. Angle(N, M, A) → υ
  80. Angle(N', M', A) → ϕ
  81. Line(E, G) → s_1
  82. Mirror(D, q) → D'_4
  83. Mirror(E, q) → E'_3
  84. Mirror(G, q) → G'_2
  85. Polygon(D'_4, E'_3, G'_2) → poly2'_2, g'_2, d'_4, e'_3
  86. Segment(E, D'_4) → f_2
  87. Segment(D'_4, G) → g_2

AI Drawing Prompt Reference

Draw a dynamic illustration of congruent triangles: a fixed triangle and a second triangle constructed to be congruent using circles, rays, and intersections based on given side ratios and angles. Include a slider that drives three transformation stages: first dilate one vertex of the constructed triangle toward the corresponding vertex of the fixed triangle, then translate the whole triangle so the vertices coincide, finally rotate the triangle about that vertex until the two triangles perfectly overlap. Show both triangles, the construction circles, rays, intersection points, and the intermediate positions during the transformation.
The prompt above was compiled from the original drawing commands and is for reference only.