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矩形相似性：通过旋转、镜像与缩放探索相似变换

矩形相似性：通过旋转、镜像与缩放探索相似变换
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2017-02-21 07:51
多边形角向量变换几何作图
本范例演示了如何通过几何变换构造并验证两个矩形的相似性。核心数学概念是相似变换，包括旋转、镜像（反射）和缩放（位似），以及多边形相似的性质。作图要点包括：使用Segment、OrthogonalLine、Point、Intersect、Angle、Polygon等命令构造原始矩形及其角度；利用Line、Mirror、Rotate、Dilate命令对矩形进行镜像、旋转和缩放，生成新的矩形；通过Angle和Polygon命令度量并比较对应角与边长比例，从而验证相似关系。该范例适合初中高年级或高中学生，用于直观理解相似图形的判定条件，以及动态几何软件中变换工具的应用。

GeoGebra 命令及参数

  1. Segment(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. OrthogonalLine(B, a) → c
  4. Point(b) → D
  5. OrthogonalLine(D, b) → d
  6. Intersect(c, d) → C
  7. Segment(D, A) → e
  8. Segment(D, C) → f
  9. Segment(C, B) → g
  10. Angle(C, B, A) → α
  11. Angle(B, A, D) → β
  12. Angle(A, D, C) → γ
  13. Angle(D, C, B) → δ
  14. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d_1
  15. Line(E, F) → h
  16. Mirror(A, h) → A'
  17. Mirror(B, h) → B'
  18. Mirror(C, h) → C'
  19. Mirror(D, h) → D'
  20. Polygon(A', B', C', D') → poly1', a', b', c', d'
  21. Point(Circle[G, Segment[B', C']]) → H
  22. Rotate(H + (Segment[C', D'], 0), Angle[D' - C'] + Angle[H - G] - Angle[C' - B'], H) → I
  23. Rotate(I + (Segment[D', A'], 0), Angle[A' - D'] + Angle[I - H] - Angle[D' - C'], I) → J
  24. Polygon(G, H, I, J) → poly3, g_1, h_1, i, j
  25. Dilate(A, 0.5, K) → A'_1
  26. Dilate(B, 0.5, K) → B'_1
  27. Dilate(C, 0.5, K) → C'_1
  28. Dilate(D, 0.5, K) → D'_1
  29. Segment(A_1, B_1) → a_2
  30. OrthogonalLine(A_1, a_2) → b_2
  31. OrthogonalLine(B_1, a_2) → c_2
  32. Point(b_2) → D_1
  33. OrthogonalLine(D_1, b_2) → d_2
  34. Intersect(c_2, d_2) → C_1
  35. Segment(D_1, A_1) → e_1
  36. Segment(D_1, C_1) → f_1
  37. Segment(C_1, B_1) → g_2
  38. Angle(C_1, B_1, A_1) → α_1
  39. Angle(B_1, A_1, D_1) → β_1
  40. Angle(A_1, D_1, C_1) → γ_1
  41. Angle(D_1, C_1, B_1) → δ_1
  42. Dilate(A_1, 0.5, K_1) → A'_2
  43. Dilate(B_1, 0.5, K_1) → B'_2
  44. Dilate(C_1, 0.5, K_1) → C'_2
  45. Dilate(D_1, 0.5, K_1) → D'_2
  46. Dilate(A, 0.5, L) → A'_3
  47. Dilate(B, 0.5, L) → B'_3
  48. Dilate(C, 0.5, L) → C'_3
  49. Dilate(D, 0.5, L) → D'_3
  50. Segment(A_2, B_2) → a_4
  51. OrthogonalLine(A_2, a_4) → b_4
  52. OrthogonalLine(B_2, a_4) → c_4
  53. Point(b_4) → D_2
  54. OrthogonalLine(D_2, b_4) → d_4
  55. Intersect(c_4, d_4) → C_2
  56. Segment(D_2, A_2) → e_2
  57. Segment(D_2, C_2) → f_2
  58. Segment(C_2, B_2) → g_3
  59. Angle(C_2, B_2, A_2) → α_2
  60. Angle(B_2, A_2, D_2) → β_2
  61. Angle(A_2, D_2, C_2) → γ_2
  62. Angle(D_2, C_2, B_2) → δ_2
  63. Dilate(A_2, 0.5, L_1) → A'_4
  64. Dilate(B_2, 0.5, L_1) → B'_4
  65. Dilate(C_2, 0.5, L_1) → C'_4
  66. Dilate(D_2, 0.5, L_1) → D'_4
  67. Polygon(A'_4, B'_4, C'_4, D'_4) → poly1'_3, a'_1, b'_1, c'_1, d'_1
  68. Dilate(A, 1.5, M) → A'_5
  69. Dilate(B, 1.5, M) → B'_5
  70. Dilate(C, 1.5, M) → C'_5
  71. Dilate(D, 1.5, M) → D'_5
  72. Rotate(A'_5, 45°, N) → O
  73. Rotate(B'_5, 45°, N) → P
  74. Rotate(C'_5, 45°, N) → Q
  75. Rotate(D'_5, 45°, N) → R
  76. Segment(A_3, B_3) → a_3
  77. OrthogonalLine(A_3, a_3) → b_3
  78. OrthogonalLine(B_3, a_3) → c_3
  79. Point(b_3) → D_3
  80. OrthogonalLine(D_3, b_3) → d_3
  81. Intersect(c_3, d_3) → C_3
  82. Segment(D_3, A_3) → e_3
  83. Segment(D_3, C_3) → f_3
  84. Segment(C_3, B_3) → g_4
  85. Angle(C_3, B_3, A_3) → α_3
  86. Angle(B_3, A_3, D_3) → β_3
  87. Angle(A_3, D_3, C_3) → γ_3
  88. Angle(D_3, C_3, B_3) → δ_3
  89. Dilate(A_3, 1.5, M_1) → A'_6
  90. Dilate(B_3, 1.5, M_1) → B'_6
  91. Dilate(C_3, 1.5, M_1) → C'_6
  92. Dilate(D_3, 1.5, M_1) → D'_6
  93. Rotate(A'_6, 45°, N_1) → O_1
  94. Rotate(B'_6, 45°, N_1) → P_1
  95. Rotate(C'_6, 45°, N_1) → Q_1
  96. Rotate(D'_6, 45°, N_1) → R_1
  97. Polygon(O_1, P_1, Q_1, R_1) → poly2_1, o_1, p_1, q_1, r_1
  98. Dilate(A, 0.75, S) → A'_7
  99. Dilate(B, 0.75, S) → B'_7
  100. Dilate(C, 0.75, S) → C'_7
  101. Dilate(D, 0.75, S) → D'_7
  102. Rotate(A'_7, 90°, T) → U
  103. Rotate(B'_7, 90°, T) → V
  104. Rotate(C'_7, 90°, T) → W
  105. Rotate(D'_7, 90°, T) → Z
  106. Segment(A_4, B_4) → a_5
  107. OrthogonalLine(A_4, a_5) → b_5
  108. OrthogonalLine(B_4, a_5) → c_5
  109. Point(b_5) → D_4
  110. OrthogonalLine(D_4, b_5) → d_5
  111. Intersect(c_5, d_5) → C_4
  112. Segment(D_4, A_4) → e_4
  113. Segment(D_4, C_4) → f_4
  114. Segment(C_4, B_4) → g_5
  115. Angle(C_4, B_4, A_4) → α_4
  116. Angle(B_4, A_4, D_4) → β_4
  117. Angle(A_4, D_4, C_4) → γ_4
  118. Angle(D_4, C_4, B_4) → δ_4
  119. Polygon(A_4, B_4, C_4, D_4) → poly1_1, a_6, b_6, c_6, d_6
  120. Dilate(A_4, 0.75, S_1) → A'_8
  121. Dilate(B_4, 0.75, S_1) → B'_8
  122. Dilate(C_4, 0.75, S_1) → C'_8
  123. Dilate(D_4, 0.75, S_1) → D'_8
  124. Polygon(A'_8, B'_8, C'_8, D'_8) → poly1'_2, a'_2, b'_2, c'_2, d'_2
  125. Rotate(A'_8, 90°, T_1) → U_1
  126. Rotate(B'_8, 90°, T_1) → V_1
  127. Rotate(C'_8, 90°, T_1) → W_1
  128. Rotate(D'_8, 90°, T_1) → Z_1
  129. Polygon(U_1, V_1, W_1, Z_1) → poly2_2, u_1, v_1, w_1, z_2
  130. Polygon(E_2, F_2, 4) → poly2_3, p_2, q_2, r_2, s_1, G_2, H_2
  131. Segment(A_5, B_5) → a_9
  132. OrthogonalLine(A_5, a_9) → b_9
  133. OrthogonalLine(B_5, a_9) → c_7
  134. Point(b_9) → D_5
  135. OrthogonalLine(G_3, b_9) → p_3
  136. Intersect(b_9, p_3) → H_3
  137. Intersect(c_7, p_3) → I_2
  138. Polygon(A_5, H_3, I_2, B_5) → poly2_4, a_8, h_3, i_2, b_8
  139. Segment(A_6, B_6) → a_{11}
  140. OrthogonalLine(A_6, a_{11}) → b_{11}
  141. OrthogonalLine(B_6, a_{11}) → c_8
  142. Point(b_{11}) → D_6
  143. OrthogonalLine(G_4, b_{11}) → p_4
  144. Intersect(b_{11}, p_4) → H_4
  145. Intersect(c_8, p_4) → I_3
  146. Polygon(A_6, H_4, I_3, B_6) → poly2_5, a_{10}, h_4, i_3, b_{10}