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Animation 231: Isosceles Right Triangles, Reflection, and Polynomial Functions

Animation 231: Isosceles Right Triangles, Reflection, and Polynomial Functions
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2017-04-06 08:05
TriangleFunctionTransformationGeometric ConstructionSlider Animation
This GeoGebra example demonstrates a dynamic animation combining isosceles right triangles, reflection transformations, and polynomial function curves. The core mathematical concepts include isosceles triangles, right triangles, reflection (mirror) symmetry, line segments, circular arcs, circles, sectors, and polynomial functions. The construction uses GeoGebra commands such as Rotate, Dilate, Mirror, Polygon, Midpoint, CircleArc, Circle, CircleSector, Point, Segment, Intersect, Ray, Line, Angle, Vector, and Translate to build a rich geometric scene. The animation likely involves sliders or dynamic parameters to show continuous transformations. It is suitable for middle or high school geometry classes, dynamic geometry exploration, or math labs, helping students visualize how reflection symmetry and triangle properties interact with function graphs.

GeoGebra Commands & Parameters

  1. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  2. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  3. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  4. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  5. Midpoint(A, C) → V
  6. CircleArc(V, C, D) → g_2
  7. Point(g_2) → B
  8. Segment(C, B) → j_1
  9. Segment(A, B) → n
  10. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  11. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  12. Segment(R, A''') → f''
  13. Intersect(n, f'') → Z
  14. Segment(B, Z) → l_1
  15. Segment(A, A') → f
  16. Segment(A'_1, A'') → f'
  17. Segment(A, B_2) → g
  18. Segment(B_2, C) → h
  19. Segment(C, D) → i
  20. Segment(D, A) → j
  21. Circle(B_2, (0.5 * Min[j_1, l_1])) → p
  22. Circle(C, (0.1 * j)) → k
  23. Circle(A, (0.1 * j)) → q
  24. Circle(D, (0.1 * j)) → r
  25. Segment(A, C) → l
  26. Segment(D, B_2) → m
  27. Intersect(r, j, 1) → F
  28. Intersect(r, m, 1) → G
  29. Intersect(r, i, 1) → H
  30. Intersect(k, i, 1) → I
  31. Intersect(k, l, 1) → J
  32. Intersect(k, h, 1) → K
  33. Intersect(p, h, 1) → L
  34. Intersect(p, m, 1) → M
  35. Intersect(p, g, 1) → N
  36. Intersect(q, g, 1) → O
  37. Intersect(q, l, 1) → P
  38. Intersect(q, j, 1) → Q
  39. CircleSector(B_2, L, N) → s
  40. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  41. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  42. Segment(R', S) → f'''
  43. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  44. Vector(B_2, B') → u
  45. Translate(s, u) → s'
  46. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  47. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  48. Vector(C, C') → v
  49. Translate(s'', v) → s'''
  50. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  51. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  52. Vector(D, D') → w
  53. Translate(t, w) → t'
  54. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  55. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  56. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  57. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  58. Vector(B_2, B'_1) → v_1
  59. Vector(B_2, B'_3) → w_1
  60. Vector(B_2, B'_2) → i_1
  61. Circle(A, n - j_1) → h_2
  62. Intersect(h_2, n, 1) → A_1
  63. Midpoint(A, A_1) → B_1
  64. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  65. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], A_1) → W'_1
  66. Segment(B, W') → m_1
  67. Segment(B, W'_1) → n_1
  68. Dilate(A_1, If[11 < e ≤ 12, 12 - e, 0], B_1) → E
  69. Segment(A_1, E) → f_1
  70. Dilate(B_1, If[12 < e ≤ 13, 13 - e, 0], A) → T
  71. Segment(B_1, T) → g_1
  72. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  73. Vector(B_2, B'_4) → u_1
  74. Translate(s, u_1) → s'_1
  75. Line(B, f') → a_1
  76. Point(a_1) → U
  77. Angle(U, B, C) → α
  78. Rotate(s'_1, If[13 < e ≤ 14, ((e - 13) * (α - π / 2)), α - π / 2], B'_4) → c_1
  79. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  80. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  81. Segment(W'_2, W'') → b_1
  82. Rotate(A_1, If[15 < e ≤ 16, ((-(e - 15)) * π), (-π)], B_1) → C_1
  83. Rotate(E, If[15 < e ≤ 16, ((-(e - 15)) * π), (-π)], B_1) → E'
  84. Segment(C_1, E') → d_1
  85. Circle(A, B_1) → e_1
  86. Line(A, B_2) → h_1
  87. Intersect(e_1, h_1, 1) → D_1
  88. Intersect(e_1, f''', 1) → E_1
  89. Ray(A, D_1) → k_1
  90. Angle(B, A, D) → β
  91. Rotate(C_1, If[16 < e ≤ 17, ((e - 16) * β), β], A) → F_1
  92. Rotate(E', If[16 < e ≤ 17, ((e - 16) * β), β], A) → E''
  93. Segment(F_1, E'') → p_1
  94. Circle(D_1, 1) → q_1
  95. Intersect(q_1, k_1, 2) → G_1
  96. Ray(D_1, G_1) → r_1
  97. Segment(A, D_1) → s_1
  98. Rotate(F_1, If[17 < e ≤ 18, ((e - 17) * π / 2), π / 2], A) → H_1
  99. Rotate(E'', If[17 < e ≤ 18, ((e - 17) * π / 2), π / 2], A) → E'''
  100. Segment(H_1, E''') → t_1
  101. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  102. Segment(D, D'_1) → f_2
  103. Rotate(t'', If[18 < e ≤ 19, ((e - 18) * π / 2), π / 2], A) → t'''
  104. Dilate(D, If[20 < e ≤ 21, 21 - e, 0], E'') → D'_2
  105. Vector(D, D'_2) → u_2
  106. Translate(D, u_2) → D'_3
  107. Translate(D'_1, u_2) → I_1
  108. Segment(D'_3, I_1) → i_2
  109. Mirror(C, D) → C'_1
  110. Line(E'', f_2) → j_2
  111. Point(j_2) → J_1
  112. Angle(J_1, E'', E''') → γ
  113. Rotate(D'_3, If[21 < e ≤ 22, ((e - 21) * γ), γ], E'') → K_1
  114. Rotate(I_1, If[21 < e ≤ 22, ((e - 21) * γ), γ], E'') → L_1
  115. Segment(K_1, L_1) → k_2
  116. Polygon(E''', A, E'') → poly2, e'', e''', a_2