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(a - b)^2 Geometric Demonstration

(a - b)^2 Geometric Demonstration
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2020-01-18 16:25
VectorCoordinate GeometryGeometric Construction
In collections Algebraic puzzles for binomial formula
This example dynamically demonstrates the geometric representation of the algebraic expression (a - b)^2 in space. It involves core math concepts like algebraic expressions, vectors, distance, rotation, and translation. GeoGebra commands and objects such as Line, Point, Polygon, Intersect, Midpoint, Vector, RotateText, Translate, Distance, and Segment are used in construction. Suitable for high school mathematics education to understand the connection between algebra and geometry and enhance spatial reasoning skills.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → a
  2. Point(a) → B
  3. Polygon(B, A, 4) → poly1, b_1, c, d, e, C, D
  4. Line(A, C) → f
  5. Point(c) → E
  6. Polygon(A, E, 4) → poly2, g, h, i, j, F, G
  7. Line(G, F) → k
  8. Line(E, F) → l
  9. Line(H, a) → m
  10. Line(I, f) → n
  11. Line(L, k) → p
  12. Line(K, d) → q
  13. Line(C, D) → r
  14. Line(B, D) → s
  15. Intersect(r, d) → M
  16. Intersect(k, r) → N
  17. Intersect(k, m) → O
  18. Intersect(m, s) → P
  19. Intersect(r, n) → Q
  20. Intersect(q, k) → R
  21. Midpoint(O, P) → S
  22. Midpoint(H, O) → T
  23. Intersect(l, p) → U
  24. Line(W, a) → t
  25. Line(V, n) → a_1
  26. Intersect(a_1, r) → Z
  27. Intersect(s, t) → A_1
  28. Midpoint(W, A_1) → B_1
  29. Vector(B_1, A_1) → u
  30. Vector(B_1, W) → v
  31. Midpoint(V, Z) → C_1
  32. Vector(C_1, V) → w
  33. Vector(C_1, Z) → z
  34. Vector(T, O) → b
  35. Vector(T, H) → u_1
  36. Midpoint(J, Q) → D_1
  37. Midpoint(I, J) → E_1
  38. Midpoint(K, R) → F_1
  39. Intersect(s, q) → G_1
  40. Vector(S, P) → v_1
  41. Vector(S, O) → w_1
  42. Vector(D_1, J) → z_1
  43. Vector(D_1, Q) → c_1
  44. Vector(F_1, R) → d_1
  45. Vector(F_1, K) → e_1
  46. Midpoint(R, G_1) → H_1
  47. RotateText(texte5, 90°) → texte7
  48. Vector(H_1, G_1) → f_1
  49. Vector(H_1, R) → g_1
  50. Vector(E_1, I) → h_1
  51. Vector(E_1, J) → i_1
  52. Midpoint(G, E) → I_1
  53. Intersect(s, e) → N_1
  54. Intersect(l, s) → O_1
  55. Polygon(G, B, O_1, F) → poly3, n_1, p_1, b_2, d_2
  56. Vector(F, K_1) → j_1
  57. Polygon(E, F, N, C) → poly4, t_1, c_2, e_2, h_2
  58. Vector(C, L_1) → k_1
  59. Polygon(F, O_1, D, N) → poly5, f_2, o_2, d_4, n_2
  60. Vector(N, M_1) → l_1
  61. Translate(G, Vector[(Animation / 10 * j_1)]) → G'
  62. Translate(B, Vector[(Animation / 10 * j_1)]) → B'
  63. Translate(D, Vector[(Animation / 10 * j_1)]) → D'
  64. Vector(C, J_1) → m_1
  65. Translate(E, Vector[(Animation / 10 * k_1)]) → E'
  66. Translate(O_1, Vector[(Animation / 10 * k_1)]) → P_1
  67. Translate(D, Vector[(Animation / 10 * k_1)]) → D'_1
  68. Translate(F, Vector[(Animation / 10 * l_1)]) → F'
  69. Translate(O_1, Vector[(Animation / 10 * l_1)]) → Q_1
  70. Translate(D, Vector[(Animation / 10 * l_1)]) → D'_2
  71. Translate(N, Vector[(Animation / 10 * l_1)]) → N'_1
  72. Polygon(F', Q_1, D'_2, N'_1) → poly5', f', q_1, d'_2, n'_1
  73. Polygon(A, G, F, E) → poly6, a_2, g_3, f_3, e_3
  74. Translate(A, Vector[(Animation / 10 * m_1)]) → A'
  75. Translate(G, Vector[(Animation / 10 * m_1)]) → G'_1
  76. Translate(F, Vector[(Animation / 10 * m_1)]) → F'_1
  77. Translate(E, Vector[(Animation / 10 * m_1)]) → E'_1
  78. Polygon(A', G'_1, F'_1, E'_1) → poly6', a', g'_1, f'_1, e'_1
  79. Polygon(A, B, D, C) → poly7, a_3, b_3, d_5, c_3
  80. Translate(A, Vector[(Animation / 10 * m_1)]) → A'_1
  81. Translate(B, Vector[(Animation / 10 * m_1)]) → B'_1
  82. Translate(D, Vector[(Animation / 10 * m_1)]) → D'_3
  83. Translate(C, Vector[(Animation / 10 * m_1)]) → C'_1
  84. Polygon(A'_1, B'_1, D'_3, C'_1) → poly7', a'_1, b'_1, d'_3, c'_1
  85. Line(E'_1, F'_1) → r_1
  86. Line(G'_1, F'_1) → s_1
  87. Intersect(r_1, b'_1) → T_1
  88. Intersect(s_1, d'_3) → U_1
  89. Translate(E, Vector[(Animation / 10 * k_1)]) → E'_2
  90. Translate(F, Vector[(Animation / 10 * k_1)]) → F'_2
  91. Translate(N, Vector[(Animation / 10 * k_1)]) → N'_2
  92. Translate(C, Vector[(Animation / 10 * k_1)]) → C'
  93. Polygon(E'_2, F'_2, N'_2, C') → poly4', e'_2, f'_2, n'_2, c'
  94. Translate(G, Vector[(Animation / 10 * j_1)]) → G'_2
  95. Translate(B, Vector[(Animation / 10 * j_1)]) → B'_2
  96. Translate(O_1, Vector[(Animation / 10 * j_1)]) → R_1
  97. Translate(F, Vector[(Animation / 10 * j_1)]) → F'_3
  98. Polygon(G'_2, B'_2, R_1, F'_3) → poly3', g'_2, b'_2, r_2, f'_3
  99. Midpoint(B', F'_3) → S_1
  100. Midpoint(E', N'_2) → V_1
  101. Midpoint(A', F'_1) → W_1
  102. Midpoint(F, A) → Z_1
  103. Midpoint(F', D'_2) → A_2
  104. Distance(F'_1, K_1) → aa
  105. Distance(C'_1, L_1) → bb
  106. Midpoint(A', D'_3) → B_2
  107. Line(G'_1, F'_1) → g_2
  108. Intersect(g_2, d'_3) → C_2
  109. Distance(M_1, C_2) → cc
  110. Intersect(d'_3, b'_1) → D_2
  111. Segment(E_2, F_2) → i_2
  112. Segment(G_2, H_2) → j_2

AI Drawing Prompt Reference

Create a dynamic geometric diagram illustrating the algebraic identity (a-b)². Start with a horizontal line and a point on it as a vertex of a large square. Construct the large square, then choose a point on one of its sides to form a smaller square inside, sharing a vertex with the large square. Draw multiple auxiliary segments connecting vertices and intersection points to create a grid-like structure. Mark midpoints of various segments and draw vector arrows from these midpoints to adjacent vertices, representing translations and distances. Finally, rotate a text label by 90 degrees and place it within the diagram to annotate the expression. The overall figure consists of squares, line segments, intersection points, and vectors, demonstrating the geometric interpretation of the algebraic formula.
The prompt above was compiled from the original drawing commands and is for reference only.