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Rectangle Similarity: Exploring Similarity via Rotation, Reflection, and Scaling

Rectangle Similarity: Exploring Similarity via Rotation, Reflection, and Scaling
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2017-02-21 07:51
PolygonAngleVectorTransformationGeometric Construction
This example demonstrates how to construct and verify the similarity of two rectangles through geometric transformations. The core mathematical concepts are similarity transformations, including rotation, reflection (mirror), and dilation (scaling), along with properties of similar polygons. Construction highlights: using commands such as Segment, OrthogonalLine, Point, Intersect, Angle, Polygon to build the original rectangle and its angles; applying Line, Mirror, Rotate, and Dilate to transform the rectangle into a new one; then using Angle and Polygon to measure and compare corresponding angles and side ratios, thereby verifying similarity. This is suitable for upper middle school or high school students to intuitively understand the criteria for similar figures and the use of transformation tools in dynamic geometry software.

GeoGebra Commands & Parameters

  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}