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Areas of Plane Figures via Geometric Transformations

Areas of Plane Figures via Geometric Transformations
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2020-08-15 17:56
PolygonVectorSolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates how to compare and/or compute the areas of plane (polygonal) figures by constructing them through geometric transformations and decomposition. The key ideas are polygon area, relationships determined by lines and midpoints, and how rotations, reflections, translations, and dilations affect area (e.g., preserving area under isometries and scaling under similarity-type transforms). Construction uses GeoGebra commands to create the underlying geometry: Point, Line, Segment, Intersect (find intersections), Midpoint (compute midpoints), OrthogonalLine (draw perpendiculars), and Polygon (build regions). It then generates transformed copies using Vector (direction for translations) together with Rotate, Translate, Mirror, and Dilate to place the figures in configurations that make area correspondence and reasoning visible. The model includes many derived points/objects (such as labeled primed objects like E', I' and related construction points) to track each transformation. Suitable for secondary school through early university geometry lessons, especially for teaching area reasoning, geometric invariants, and transformation-based problem solving.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. OrthogonalLine(A, f) → g
  3. OrthogonalLine(B, f) → h
  4. Point(h) → C
  5. Line(C, f) → i
  6. Intersect(g, i) → D
  7. Midpoint(E, A) → F
  8. Midpoint(B, E) → G
  9. OrthogonalLine(E, f) → j
  10. Intersect(f, j) → H
  11. Line(F, f) → k
  12. Intersect(j, k) → I
  13. Polygon(E, F, I) → t1, i_1, e, f_1
  14. Polygon(I, G, E) → t2, e_1, i_2, g_1
  15. Polygon(F, A, B, G) → q1, f_2, a, b, g_2
  16. Rotate(E, (π * c / 2), F) → E'
  17. Rotate(F, (π * c / 2), F) → F'
  18. Rotate(I, (π * c / 2), F) → I'
  19. Polygon(E', F', I') → t1', i', e', f'
  20. Rotate(I, ((-π) * c / 2), G) → I'_1
  21. Rotate(G, ((-π) * c / 2), G) → G'
  22. Rotate(E, ((-π) * c / 2), G) → E'_1
  23. Polygon(I'_1, G', E'_1) → t2', e'_1, i'_1, g'
  24. Segment(D, C) → l
  25. Point(f) → J
  26. Segment(C, J) → m
  27. Line(A, m) → n
  28. Intersect(l, n) → K
  29. OrthogonalLine(K, f) → p
  30. Intersect(p, f) → L
  31. Polygon(K, A, L) → t3, l_1, k_1, a_1
  32. Polygon(L, J, C, K) → q2, l_2, j_1, c_1, k_2
  33. Vector(A, V) → u
  34. Translate(K, u) → K'
  35. Translate(A, u) → A'
  36. Translate(L, u) → L'
  37. Polygon(K', A', L') → t3', l', k', a'
  38. Point(l) → M
  39. Point(l) → N
  40. Midpoint(M, A) → O
  41. Midpoint(N, B) → P
  42. OrthogonalLine(P, f) → q
  43. OrthogonalLine(O, f) → r
  44. Intersect(r, f) → Q
  45. Intersect(q, f) → R
  46. Polygon(M, O, Q, R, P, N) → poly1, m_1, o_1, q_1, r_1, p_1, n_1
  47. Polygon(O, A, Q) → t4, q_2, o_2, a_2
  48. Polygon(R, B, P) → t5, p_2, r_2, b_1
  49. Rotate(R, (π * c / 2), P) → R'
  50. Rotate(B, (π * c / 2), P) → B'
  51. Rotate(P, (π * c / 2), P) → P'
  52. Polygon(R', B', P') → t5', p', r', b'
  53. Rotate(O, ((-π) * c / 2), O) → O'
  54. Rotate(A, ((-π) * c / 2), O) → A'_1
  55. Rotate(Q, ((-π) * c / 2), O) → Q'
  56. Polygon(O', A'_1, Q') → t4', q', o', a'_1
  57. Mirror(A, D) → A'_2
  58. Mirror(B, C) → B'_1
  59. Midpoint(A'_2, B'_1) → S
  60. Midpoint(A, B) → T
  61. Midpoint(D, C) → U
  62. Polygon(S, D, U) → t6, u_1, s_1, d_1
  63. Polygon(S, U, C) → t7, c_2, s_2, u_2
  64. Polygon(D, T, U) → t8, u_3, d_2, t
  65. Polygon(T, C, U) → t9, u_4, t_1, c_3
  66. Rotate(S, ((π * 3) * c / 4), D) → S'
  67. Rotate(D, ((π * 3) * c / 4), D) → D'
  68. Rotate(U, ((π * 3) * c / 4), D) → U'
  69. Rotate(S, (((-π) * 3) * c / 4), C) → S'_1
  70. Rotate(U, (((-π) * 3) * c / 4), C) → U'_1
  71. Rotate(C, (((-π) * 3) * c / 4), C) → C'
  72. Midpoint(A'_2, D) → W
  73. Midpoint(B'_1, C) → Z
  74. Midpoint(T, Z) → A_1
  75. Midpoint(T, W) → B_1
  76. Midpoint(S, W) → C_1
  77. Midpoint(S, Z) → D_1
  78. Segment(S, T) → h_1
  79. Segment(C_1, D_1) → h_2
  80. Segment(B_1, A_1) → j_2
  81. Intersect(j_2, h_1) → E_1
  82. Intersect(h_1, h_2) → F_1
  83. Polygon(C_1, F_1, S) → t10, s_3, m_2, n_2
  84. Polygon(F_1, D_1, S) → t11, s_4, t_2, b_2
  85. Polygon(T, A_1, E_1) → t12, e_2, t_3, f_3
  86. Polygon(E_1, B_1, T) → t13, t_4, g_3, h_3
  87. Segment(W, Z) → i_3
  88. Intersect(i_3, h_1) → G_1
  89. Polygon(F_1, G_1, Z, D_1) → q3, j_3, k_3, z_1, l_3
  90. Polygon(C_1, W, G_1, F_1) → q4, m_3, w, n_3, p_3
  91. Polygon(E_1, A_1, Z, G_1) → q5, q_3, r_3, z_2, a_3
  92. Polygon(B_1, E_1, G_1, W) → q6, b_3, d_3, e_3, w_1
  93. Rotate(C_1, (π * c / 2), C_1) → H_1
  94. Rotate(F_1, (π * c / 2), C_1) → I_1
  95. Rotate(S, (π * c / 2), C_1) → S'_2
  96. Polygon(H_1, I_1, S'_2) → t10', s'_2, f_4, g_4
  97. Rotate(F_1, ((-π) * c / 2), D_1) → J_1
  98. Rotate(D_1, ((-π) * c / 2), D_1) → K_1
  99. Rotate(S, ((-π) * c / 2), D_1) → S'_3
  100. Polygon(J_1, K_1, S'_3) → t11', s'_3, h_4, i_4
  101. Rotate(T, (π * c / 2), A_1) → T'
  102. Rotate(A_1, (π * c / 2), A_1) → L_1
  103. Rotate(E_1, (π * c / 2), A_1) → M_1
  104. Polygon(T', L_1, M_1) → t12', j_4, t', k_4
  105. Rotate(E_1, ((-π) * c / 2), B_1) → N_1
  106. Rotate(B_1, ((-π) * c / 2), B_1) → O_1
  107. Rotate(T, ((-π) * c / 2), B_1) → T'_1
  108. Polygon(N_1, O_1, T'_1) → t13', t'_1, l_4, m_4
  109. Midpoint(D, S) → R_1
  110. Midpoint(T, C) → S_1
  111. Midpoint(S, C) → T_1
  112. Midpoint(D, T) → U_1
  113. Dilate(R_1, If[0 < c ≤ 1, 1 - c, 0], S_1) → V_1
  114. Vector(R_1, V_1) → v_2
  115. Dilate(T_1, If[1 < c ≤ 2, 2 - c, 0], U_1) → W_1
  116. Vector(T_1, W_1) → w_2
  117. Translate(S, v_2) → S'_4
  118. Translate(D, v_2) → D'_1
  119. Translate(U, v_2) → U'_2
  120. Polygon(S'_4, D'_1, U'_2) → t6', u'_2, s'_4, d'_1
  121. Translate(S, w_2) → S'_5
  122. Translate(U, w_2) → U'_3
  123. Translate(C, w_2) → C'_1
  124. Polygon(S'_5, U'_3, C'_1) → t7', c'_1, s'_5, u'_3

AI Drawing Prompt Reference

Draw a graphic composed of multiple planar polygons to demonstrate how geometric transformations such as rotation, reflection, translation, and dilation affect area. The graphic should include original polygons and their transformed copies, which are placed via rotation and translation to show area correspondence. Key objects include points, line segments, and polygons, connected through intersections, perpendicular lines, and midpoints. The graphic should clearly illustrate the correspondence between transformed and original polygons, suitable for geometry teaching and area reasoning.
The prompt above was compiled from the original drawing commands and is for reference only.