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Geometric Construction for the Area of a Rectangle

Geometric Construction for the Area of a Rectangle
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2023-03-13 03:56
CirclePolygonGeometric Construction
In collections geometriadinamica.es
This GeoGebra example demonstrates how to construct a rectangle using perpendicular and intersection relationships, then verify its area using the rectangle’s side measures (base and height). The key math concepts are rectangle properties (opposite sides parallel, adjacent sides perpendicular), constructing perpendicular lines, locating vertices via intersections, and using segment lengths as geometric quantities. Construction highlights: it employs GeoGebra commands Circle, Ray, OrthogonalLine, Intersect, Polygon, Mirror, Segment, and Point to build the configuration. From the provided construction summary, it uses objects such as base and altura (height), and multiple labeled points (e.g., a_1, c_1, e_1, d_1) to define key vertices. OrthogonalLine establishes the height direction, Intersect determines corner points, Polygon closes the region into a rectangle, Segment provides the base/height measurements for the area relation, and Mirror generates corresponding structures to support geometric validation. Suitable for middle-school geometry instruction, interactive construction practice, and teacher-led demonstrations on why area equals base times height.

GeoGebra Commands & Parameters

  1. Circle(A, base) → c
  2. Ray(A, B) → a
  3. Circle(A, altura) → d
  4. OrthogonalLine(A, a) → b
  5. Intersect(d, b, 1) → C
  6. OrthogonalLine(C, b) → e
  7. Intersect(c, a, 1) → D
  8. OrthogonalLine(D, a) → f
  9. Intersect(f, e) → E
  10. Polygon(A, C, E, D) → P, a_1, c_1, e_1, d_1
  11. Circle(A, g) → h
  12. Intersect(h, a_1) → F,
  13. Intersect(h, d_1) → G,
  14. Mirror(A, G) → H
  15. Mirror(G, H) → I
  16. Mirror(H, I) → J
  17. Mirror(I, J) → K
  18. Mirror(J, K) → L
  19. Mirror(K, L) → M
  20. Mirror(L, M) → N
  21. Mirror(M, N) → O
  22. Mirror(N, O) → Q
  23. Mirror(O, Q) → R
  24. Mirror(Q, R) → S
  25. Mirror(R, S) → T
  26. Mirror(S, T) → U
  27. Mirror(A, F) → V
  28. Mirror(F, V) → W
  29. Mirror(V, W) → Y
  30. Mirror(W, Y) → Z
  31. Mirror(Y, Z) → A_1
  32. Mirror(Z, A_1) → B_1
  33. Mirror(A_1, B_1) → C_1
  34. Mirror(B_1, C_1) → D_1
  35. OrthogonalLine(G, d_1) → i
  36. OrthogonalLine(H, d_1) → j
  37. OrthogonalLine(I, d_1) → k
  38. OrthogonalLine(J, d_1) → l
  39. OrthogonalLine(K, d_1) → m
  40. OrthogonalLine(L, d_1) → n
  41. OrthogonalLine(M, d_1) → p
  42. OrthogonalLine(N, d_1) → q
  43. OrthogonalLine(O, d_1) → r
  44. OrthogonalLine(Q, d_1) → s
  45. OrthogonalLine(R, d_1) → t
  46. OrthogonalLine(S, d_1) → b_1
  47. OrthogonalLine(T, d_1) → f_1
  48. OrthogonalLine(U, d_1) → g_1
  49. OrthogonalLine(F, a_1) → h_1
  50. OrthogonalLine(V, a_1) → i_1
  51. OrthogonalLine(W, a_1) → j_1
  52. OrthogonalLine(Y, a_1) → k_1
  53. OrthogonalLine(Z, a_1) → l_1
  54. OrthogonalLine(A_1, a_1) → m_1
  55. OrthogonalLine(B_1, a_1) → n_1
  56. OrthogonalLine(C_1, a_1) → p_1
  57. OrthogonalLine(D_1, a_1) → q_1
  58. Intersect(c_1, i) → E_1
  59. Intersect(r, c_1) → F_1
  60. Segment(F_1, O) → r_1
  61. Intersect(j, c_1) → G_1
  62. Intersect(k, c_1) → H_1
  63. Intersect(l, c_1) → I_1
  64. Intersect(m, c_1) → J_1
  65. Intersect(n, c_1) → K_1
  66. Intersect(p, c_1) → L_1
  67. Intersect(q, c_1) → M_1
  68. Intersect(s, c_1) → N_1
  69. Intersect(t, c_1) → O_1
  70. Intersect(b_1, c_1) → P_1
  71. Intersect(f_1, c_1) → Q_1
  72. Intersect(g_1, c_1) → R_1
  73. Intersect(e_1, q_1) → S_1
  74. Intersect(e_1, p_1) → T_1
  75. Intersect(e_1, n_1) → U_1
  76. Intersect(m_1, e_1) → V_1
  77. Intersect(l_1, e_1) → W_1
  78. Intersect(k_1, e_1) → Y_1
  79. Intersect(j_1, e_1) → Z_1
  80. Intersect(i_1, e_1) → A_2
  81. Intersect(h_1, e_1) → B_2
  82. Segment(E_1, G) → s_1
  83. Segment(G_1, H) → t_1
  84. Segment(H_1, I) → a_2
  85. Segment(I_1, J) → b_2
  86. Segment(J_1, K) → c_2
  87. Segment(K_1, L) → d_2
  88. Segment(L_1, M) → e_2
  89. Segment(M_1, N) → f_2
  90. Segment(N_1, Q) → g_2
  91. Segment(O_1, R) → h_2
  92. Segment(P_1, S) → i_2
  93. Segment(Q_1, T) → j_2
  94. Segment(R_1, U) → k_2
  95. Segment(D_1, S_1) → l_2
  96. Segment(C_1, T_1) → m_2
  97. Segment(B_1, U_1) → n_2
  98. Segment(A_1, V_1) → p_2
  99. Segment(Z, W_1) → q_2
  100. Segment(Y, Y_1) → r_2
  101. Segment(W, Z_1) → s_2
  102. Segment(V, A_2) → t_2
  103. Segment(F, B_2) → a_3
  104. Circle(C_2, o) → c_3
  105. Point(c_3) → D_2
  106. Segment(C_2, D_2) → b_3
  107. Circle(D_2, C_2) → d_3
  108. OrthogonalLine(D_2, b_3) → e_3
  109. OrthogonalLine(C_2, b_3) → f_3
  110. Intersect(c_3, f_3, 2) → E_2
  111. Intersect(d_3, e_3, 2) → F_2
  112. Polygon(E_2, F_2, D_2, C_2) → P_2, e_4, f_4, d_4, c_4

AI Drawing Prompt Reference

Draw a rectangle with a horizontal base ray and a vertical height ray. Start from the base endpoint, construct a perpendicular line to the base, intersecting the height circle to get the first vertex. From this vertex, draw another perpendicular line to intersect the base circle, obtaining the second vertex. The intersection of these two perpendicular lines gives the third vertex. Connect the four vertices to form the rectangle. Then, using one rectangle vertex as center, draw a circle with a given radius, intersecting two sides of the rectangle at two points. From these points, construct a series of perpendicular lines along the rectangle sides, and use mirror reflections to create a symmetric grid inside the rectangle, illustrating that the area equals base times height.
The prompt above was compiled from the original drawing commands and is for reference only.