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Area of Quadrilaterals: Dynamic Construction of Parallelogram, Rectangle, Rhombu

Area of Quadrilaterals: Dynamic Construction of Parallelogram, Rectangle, Rhombu
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2014-06-07 08:52
PolygonAngleCoordinate GeometryTransformationGeometric Construction
In collections Deriving area formulas for Geometry Surface nets 6th Grade Deriving area formulas for Geometry
This GeoGebra example dynamically demonstrates the area relationships and construction of various quadrilaterals, including parallelograms, rectangles, rhombi, and trapezoids. The core mathematical concepts involve definitions of quadrilaterals, area formulas, symmetry, rotation and reflection transformations, and set-theoretic classification of shapes. The construction uses commands such as Line, OrthogonalLine, Point, Intersect, Segment, Midpoint, Semicircle, Rotate, Angle, Polygon, Mirror, and Circle, creating multiple points, segments, polygons, and auxiliary circles. Through rotation and mirroring, symmetric figures are generated, resulting in two polygons (poly1 and poly2) for area comparison. This resource is suitable for middle and high school geometry classes, helping students intuitively understand the derivation of area formulas for different quadrilaterals and the effects of transformations on area, as well as exploring area conservation and equidecomposition problems.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. Point(b) → C
  4. Line(C, a) → c
  5. OrthogonalLine(B, a) → d
  6. Intersect(c, d) → D
  7. Segment(A, B) → e
  8. Point(e) → E
  9. Segment(E, C) → f
  10. Segment(E, D) → g
  11. Midpoint(f) → F
  12. Midpoint(g) → G
  13. Semicircle(E, D) → h
  14. OrthogonalLine(E, e) → i
  15. Intersect(c, i) → H
  16. Semicircle(B, H) → k
  17. Semicircle(D, E) → p_1
  18. Semicircle(H, A) → q
  19. Semicircle(C, E) → r
  20. Semicircle(E, C) → s
  21. Rotate(E, -α, G) → E'
  22. Angle(E', G, E) → β
  23. Rotate(B, -α, G) → B'
  24. Angle(B', G, B) → γ
  25. Rotate(D, -α, G) → D'
  26. Angle(D', G, D) → δ
  27. Polygon(E', B', D') → poly1, d', e', b'
  28. Rotate(A, α, F) → A'
  29. Angle(A, F, A') → ε
  30. Rotate(E, α, F) → E'_1
  31. Angle(E, F, E'_1) → ζ
  32. Rotate(C, α, F) → C'
  33. Angle(C, F, C') → η
  34. Polygon(A', C', E'_1) → poly2, e'_1, a', c'
  35. Midpoint(e) → I
  36. Midpoint(A, C) → J
  37. Midpoint(C, D) → K
  38. Midpoint(B, D) → L
  39. Polygon(J, I, L, K) → poly3, j_1, i_1, l, k_1
  40. Midpoint(j_1) → M
  41. Midpoint(i_1) → N
  42. Midpoint(l) → O
  43. Midpoint(k_1) → P
  44. Rotate(J, α, P) → J'
  45. Angle(J, P, J') → θ
  46. Rotate(C, α, P) → C'_1
  47. Angle(C, P, C'_1) → ι
  48. Rotate(K, α, P) → K'
  49. Angle(K, P, K') → κ
  50. Polygon(J', C'_1, K') → poly4, k', j', c'_1
  51. Rotate(K, -α, O) → K'_1
  52. Angle(K'_1, O, K) → λ
  53. Rotate(L, -α, O) → L'
  54. Angle(L', O, L) → μ
  55. Rotate(D, -α, O) → D'_1
  56. Angle(D'_1, O, D) → ν
  57. Polygon(K'_1, D'_1, L') → poly5, l', k'_1, d'_1
  58. Rotate(A, -α, M) → A'_1
  59. Angle(A'_1, M, A) → ξ
  60. Rotate(I, -α, M) → I'
  61. Angle(I', M, I) → ο
  62. Rotate(J, -α, M) → J'_1
  63. Angle(J'_1, M, J) → ρ
  64. Polygon(J'_1, A'_1, I') → poly6, i', j'_1, a'_1
  65. Rotate(B, α, N) → B'_1
  66. Angle(B, N, B'_1) → σ
  67. Rotate(L, α, N) → L'_1
  68. Angle(L, N, L'_1) → τ
  69. Rotate(I, α, N) → I'_1
  70. Angle(I, N, I'_1) → υ
  71. Polygon(B'_1, L'_1, I'_1) → poly7, i'_1, b'_1, l'_1
  72. Segment(A', C') → n_2
  73. Polygon(C, E, D) → poly8, d_1, c_1, e_1
  74. Midpoint(A, C) → Q
  75. Midpoint(D, C) → R
  76. Midpoint(A, B) → S
  77. Midpoint(B, D) → T
  78. Segment(Q, T) → t
  79. Segment(S, R) → a_1
  80. Intersect(t, a_1) → U
  81. Point(e) → V
  82. Mirror(V, U) → V'
  83. Polygon(C, Q, V, B, T, V') → poly9, c_2, q_1, v, b_1, t_1, v'
  84. Rotate(A, α, Q) → A'_2
  85. Angle(A, Q, A'_2) → ϕ
  86. Rotate(V, α, Q) → V'_1
  87. Angle(V, Q, V'_1) → χ
  88. Polygon(Q, V'_1, A'_2) → poly10, a'_2, q_2, v'_1
  89. Rotate(V', α, T) → V''
  90. Angle(V', T, V'') → ψ
  91. Rotate(D, α, T) → D'_2
  92. Angle(D, T, D'_2) → ω
  93. Polygon(V'', D'_2, T) → poly11, t_2, v'', d'_2
  94. Segment(C, D) → f_1
  95. Segment(C, V') → g_1
  96. Segment(V, B) → h_1
  97. Segment(Q, C) → l_1
  98. Midpoint(A, C) → W
  99. Midpoint(B, D) → Z
  100. Segment(V, B) → m_1
  101. Point(m_1) → A_1
  102. Polygon(W, C, D, Z, A_1, V) → poly12, w, c_3, d_2, z, u_2, v_1
  103. Rotate(A_1, -α, Z) → B_1
  104. Angle(B_1, Z, A_1) → α_1
  105. Rotate(B, -α, Z) → B'_2
  106. Angle(B'_2, Z, B) → β_1
  107. Polygon(B_1, Z, B'_2) → poly13, b'_2, b_2, z_1
  108. Segment(E, C) → n_1
  109. Segment(C, D) → r_1
  110. Segment(E, D) → s_1
  111. Midpoint(s_1) → C_1
  112. Rotate(E, α, C_1) → E'_2
  113. Angle(E, C_1, E'_2) → γ_1
  114. Rotate(C, α, C_1) → C'_2
  115. Angle(C, C_1, C'_2) → δ_1
  116. Rotate(D, α, C_1) → D'_3
  117. Angle(D, C_1, D'_3) → ε_1
  118. Polygon(E'_2, C'_2, D'_3) → poly14, d'_3, e'_2, c'_2
  119. Midpoint(A, C) → D_1
  120. Midpoint(B, D) → E_1
  121. Mirror(V, D_1) → V'_2
  122. Mirror(A_1, E_1) → F_1
  123. Polygon(V, V'_2, F_1, A_1) → poly15, v_2, v'_2, f_2, a_2
  124. Midpoint(A_1, F_1) → G_1
  125. Rotate(A_1, α, G_1) → H_1
  126. Angle(A_1, G_1, H_1) → ζ_1
  127. Rotate(V, α, G_1) → V'_3
  128. Angle(V, G_1, V'_3) → η_1
  129. Rotate(V'_2, α, G_1) → I_1
  130. Angle(V'_2, G_1, I_1) → θ_1
  131. Rotate(F_1, α, G_1) → J_1
  132. Angle(F_1, G_1, J_1) → ι_1
  133. Polygon(V'_3, I_1, J_1, H_1) → poly16, v'_3, i_2, j_2, h_2
  134. Polygon(E'_1, A', C') → poly17, c'_3, e'_3, a'_3
  135. Polygon(A, C, D, B) → poly18, a_3, c_4, d_3, b_3
  136. Polygon(C, E, D) → poly19, d_4, c_5, e_2
  137. Polygon(E', B', D') → poly20, d'_4, e'_4, b'_3
  138. Polygon(A'_1, J'_1, I') → poly21, i'_2, a'_4, j'_2
  139. Polygon(B'_1, I'_1, L'_1) → poly22, l'_2, b'_4, i'_3
  140. Polygon(J', K', C'_1) → poly23, c'_4, j'_3, k'_2
  141. Polygon(K'_1, D'_1, L') → poly24, l'_3, k'_3, d'_5
  142. Polygon(A, C, D, B) → poly25, a_4, c_6, d_5, b_4
  143. Polygon(I, J, K, L) → poly26, i_3, j_3, k_2, l_2
  144. Polygon(V'_1, A'_2, Q) → poly27, q_3, v'_4, a'_5
  145. Polygon(D'_2, T, V'') → poly28, v''_1, d'_6, t_3
  146. Polygon(V, V'_1, V', V'') → poly29, v_3, v'_5, v'_6, v''_2
  147. Polygon(A, C, D, B) → poly30, a_5, c_7, d_6, b_5
  148. Point(b_5) → K_1
  149. Polygon(V'_1, A'_2, Q) → poly31, q_4, v'_7, a'_6
  150. Polygon(B_1, Z, B'_2) → poly32, b'_5, b_6, z_2
  151. Polygon(V, V'_1, B_1, A_1) → poly33, v_4, v'_8, b_7, a_6
  152. Polygon(A, C, D, B) → poly34, a_7, c_8, d_7, b_8
  153. Polygon(E, C, D) → poly35, d_8, e_3, c_9
  154. Polygon(D'_3, E'_2, C'_2) → poly36, c'_5, d'_7, e'_5
  155. Polygon(C'_2, E, C, D) → poly37, c'_6, e_4, c_{10}, d_9
  156. Polygon(V, V'_2, F_1, A_1) → poly38, v_5, v'_9, f_3, a_8
  157. Polygon(J_1, H_1, V'_3, I_1) → poly39, j_4, h_3, v'_{10}, i_4
  158. Polygon(V, V'_2, V'_3, I_1) → poly40, v_6, v'_{11}, v'_{12}, i_5
  159. Midpoint(A, E) → L_1
  160. Midpoint(E, B) → M_1
  161. Mirror(C, L_1) → C'_3
  162. Polygon(A, C, D, B) → poly41, a_9, c_{11}, d_{10}, b_9
  163. Polygon(L_1, C, D, M_1) → poly42, l_3, c_{12}, d_{11}, m_2
  164. Mirror(A, L_1) → A'_3
  165. Mirror(A'_3, L_1) → O_1
  166. Intersect(l_3, c_{12}) → P_1
  167. Rotate(O_1, -α, L_1) → Q_1
  168. Angle(Q_1, L_1, O_1) → κ_1
  169. Rotate(P_1, -α, L_1) → R_1
  170. Angle(R_1, L_1, P_1) → λ_1
  171. Polygon(Q_1, R_1, L_1) → poly43, l_4, q_5, r_2
  172. Rotate(D, α, M_1) → D'_4
  173. Angle(D, M_1, D'_4) → μ_1
  174. Rotate(B, α, M_1) → B'_3
  175. Angle(B, M_1, B'_3) → ν_1
  176. Polygon(M_1, B'_3, D'_4) → poly44, d'_8, m_3, b'_6
  177. Polygon(R_1, C, D) → poly45, d_{12}, r_3, c_{13}
  178. Circle(C, D) → p_2
  179. Line(A, B) → g_2
  180. Intersect(p_2, g_2) → N_1, S_1
  181. Polygon(A, C, D, B) → poly46, a_{10}, c_{14}, d_{13}, b_{10}
  182. Polygon(C, S_1, B, D) → poly47, c_{15}, s_2, b_{11}, d_{14}
  183. Segment(A, B) → g_3
  184. Midpoint(g_3) → T_1
  185. Rotate(A, -α, T_1) → A'_4
  186. Angle(A'_4, T_1, A) → ξ_1
  187. Segment(C, D) → k_3
  188. Midpoint(c_{14}) → U_1
  189. Rotate(C, -α, U_1) → C'_4
  190. Angle(C'_4, U_1, C) → ο_1
  191. Midpoint(S_1, B) → V_1
  192. Mirror(A, V_1) → A'_5
  193. Midpoint(S_1, A'_5) → W_1
  194. Rotate(S_1, -α, W_1) → Z_1
  195. Angle(Z_1, W_1, S_1) → ρ_1
  196. Polygon(A'_4, C'_4, Z_1) → poly48, z_3, a'_7, c'_7
  197. Polygon(C, S_1, Z_1, D) → poly49, c_{16}, s_3, z_4, d_{15}

AI Drawing Prompt Reference

Draw a dynamic geometric illustration showing area relationships and constructions of parallelograms, rectangles, rhombuses, and trapezoids. Start by drawing a line segment and its perpendicular, forming a right triangle. Use the segment as a diameter to draw a semicircle, connecting the intersections to create a parallelogram. Rotate the semicircle around the midpoint of the segment to form another parallelogram for area comparison. Connect various points to form multiple polygons, and rotate some to demonstrate symmetry and area changes.
The prompt above was compiled from the original drawing commands and is for reference only.