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相似图形面积变化探究

相似图形面积变化探究
几何画板及绘图内容加载中，请稍等……
2018-05-20 09:30
几何作图
所属专题 Perimeter & Area
本范例演示相似图形在比例因子k缩放时面积的变化规律。涉及核心数学概念包括相似变换、面积比例关系。作图要点运用了Dilate（缩放）命令来创建缩放后的图形，结合Polygon（多边形）对象展示面积变化。适合高中阶段学生用于理解相似图形性质及比例关系，可作为几何作图与变换教学的辅助工具。

GeoGebra 命令及参数

  1. Rotate(B, (a * π / 180), A) → B'
  2. Angle(B, A, B') → α
  3. Segment(A, B) → f
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Segment(A, C) → h
  7. Segment(C, B) → i
  8. Line(A, C) → j
  9. Line(C, B) → k
  10. Line(A, B) → l
  11. Circle(C, A) → c
  12. Intersect(c, j, 2) → D
  13. Circle(D, C) → d
  14. Intersect(d, j, 2) → E
  15. Line(D, i) → m
  16. Line(E, i) → n
  17. Intersect(l, m) → F
  18. Intersect(l, n) → G
  19. Line(C, l) → p
  20. Line(D, l) → q
  21. Intersect(m, p) → H
  22. Intersect(n, q) → I
  23. Line(B, H) → r
  24. Line(F, r) → s
  25. Polygon(A, B, C) → t1, c_1, a_1, b
  26. Rotate(A, If[0 < e ≤ 1, (e * π), π], C) → A'
  27. Segment(C, A') → t
  28. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π), (-π)], B) → A'_1
  29. Segment(B, A'_1) → f_1
  30. Dilate(C, If[2 < e ≤ 3, 3 - e, 0], D) → C'
  31. Vector(C, C') → u
  32. Translate(C, u) → C'_1
  33. Translate(B, u) → B'_1
  34. Segment(C'_1, B'_1) → i'
  35. Dilate(B, If[3 < e ≤ 4, 4 - e, 0], F) → C'_2
  36. Vector(B, C'_2) → v
  37. Translate(C, v) → C'_3
  38. Translate(B, v) → B'_2
  39. Segment(C'_3, B'_2) → i'_1
  40. Dilate(C, If[4 < e ≤ 5, 5 - e, 0], H) → C'_4
  41. Segment(C, C'_4) → g_1
  42. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], H) → B'_3
  43. Segment(B, B'_3) → h_1
  44. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'_5
  45. Vector(C, C'_5) → w
  46. Translate(A, w) → A'_2
  47. Translate(B, w) → B'_4
  48. Translate(C, w) → C'_6
  49. Polygon(A'_2, B'_4, C'_6) → t1', c'_6, a'_2, b'_4
  50. Dilate(D, If[7.75 < e ≤ 8.75, 8.75 - e, 0], H) → D'
  51. Vector(D, D') → u_1
  52. Translate(A'_2, u_1) → J
  53. Translate(B'_4, u_1) → K
  54. Translate(C'_6, u_1) → L
  55. Polygon(J, K, L) → t1'', l_1, j_1, k_1
  56. Midpoint(h_1) → M
  57. Rotate(J, If[9.5 < e ≤ 10.5, ((e - 9.5) * π), π], M) → J'
  58. Rotate(K, If[9.5 < e ≤ 10.5, ((e - 9.5) * π), π], M) → K'
  59. Rotate(L, If[9.5 < e ≤ 10.5, ((e - 9.5) * π), π], M) → L'
  60. Polygon(J', K', L') → t1''', l', j', k'
  61. Rotate(C, If[13 < e ≤ 14, ((e - 13) * π), π], D) → C'_7
  62. Segment(D, C'_7) → i_1
  63. Midpoint(I, G) → N
  64. Rotate(B, If[13 < e ≤ 14, ((-(e - 13)) * π), (-π)], F) → B'_5
  65. Segment(F, B'_5) → m_1
  66. Dilate(D, If[14 < e ≤ 15, 15 - e, 0], E) → D'_1
  67. Vector(D, D'_1) → v_1
  68. Translate(C'_1, v_1) → O
  69. Translate(B'_1, v_1) → P
  70. Segment(O, P) → i''
  71. Translate(C'_3, v_1) → Q
  72. Translate(B'_2, v_1) → R
  73. Segment(Q, R) → n_1
  74. Dilate(F, If[15 < e ≤ 16, 16 - e, 0], G) → F'
  75. Vector(F, F') → w_1
  76. Translate(C'_3, w_1) → S
  77. Translate(B'_2, w_1) → T
  78. Segment(S, T) → p_1
  79. Dilate(H, If[16 < e ≤ 17, 17 - e, 0], N) → H'
  80. Dilate(H, If[16 < e ≤ 17, 17 - e, 0], I) → H'_1
  81. Dilate(D, If[16 < e ≤ 17, 17 - e, 0], I) → D'_2
  82. Dilate(F, If[16 < e ≤ 17, 17 - e, 0], N) → F'_1
  83. Segment(F, F'_1) → q_1
  84. Segment(H, H') → r_1
  85. Segment(H, H'_1) → s_1
  86. Segment(D, D'_2) → t_1
  87. Dilate(D, If[17.75 < e ≤ 18.75, 18.75 - e, 0], E) → D'_3
  88. Vector(D, D'_3) → b_1
  89. Dilate(F, If[17.75 < e ≤ 18.75, 18.75 - e, 0], G) → F'_2
  90. Vector(F, F'_2) → d_1
  91. Translate(A'_2, b_1) → U
  92. Translate(B'_4, b_1) → V
  93. Translate(C'_6, b_1) → W
  94. Polygon(U, V, W) → t1''_1, w_2, u_2, v_2
  95. Translate(J, d_1) → J'_1
  96. Translate(K, d_1) → K'_1
  97. Translate(L, d_1) → L'_1
  98. Polygon(J'_1, K'_1, L'_1) → t1'''_1, l'_1, j'_1, k'_1
  99. Dilate(N, If[19.5 < e ≤ 20.5, 20.5 - e, 0], I) → N'
  100. Vector(N, N') → e_1
  101. Translate(J'_1, e_1) → Z
  102. Translate(K'_1, e_1) → A_1
  103. Translate(L'_1, e_1) → B_1
  104. Polygon(Z, A_1, B_1) → t2, b_2, z_1, a_2
  105. Midpoint(r_1) → C_1
  106. Rotate(Z, If[21.25 < e ≤ 22.25, ((e - 21.25) * π), π], C_1) → Z'
  107. Rotate(A_1, If[21.25 < e ≤ 22.25, ((e - 21.25) * π), π], C_1) → D_1
  108. Rotate(B_1, If[21.25 < e ≤ 22.25, ((e - 21.25) * π), π], C_1) → E_1
  109. Polygon(Z', D_1, E_1) → t2', e_2, z', d_2
  110. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], D) → C'_8
  111. Vector(C, C'_8) → c_2
  112. Translate(J', c_2) → J''
  113. Translate(K', c_2) → K''
  114. Translate(L', c_2) → L''
  115. Polygon(J'', K'', L'') → t3, l'', j'', k''