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Geometric Area Reflection Demo

Geometric Area Reflection Demo
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2020-10-22 05:14
VectorTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates exploring the change in geometric area through reflection transformations. It involves core math concepts like area calculation, reflection, analytical geometry, and trigonometry. Construction highlights include commands/objects like Line, Point, Vector, Translate, OrthogonalLine, Intersect, Segment, Polygon, Midpoint, Circle, Mirror, Semicircle, Angle, Rotate, CircumcircleArc, Sequence, Min, Max from GeoGebra. Suitable for high school and above for learning geometric transformations and area calculations visually.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → b_3
  2. Point(b_3) → B
  3. Vector(A, O) → e
  4. Translate(B, e) → B'_4
  5. Line(A, B) → a_6
  6. OrthogonalLine(B, a_6) → c
  7. Translate(C, e) → C'_4
  8. OrthogonalLine(C, c) → d
  9. OrthogonalLine(A, a_6) → b
  10. Intersect(d, b) → D
  11. Translate(D, e) → D'_4
  12. Segment(E, E + (2, 0)) → a
  13. Vector(A, E_2) → u
  14. Vector(A, F_2) → v
  15. Vector(A, G) → w
  16. Vector(A, H) → z
  17. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d_1
  18. Translate(A, u) → A'
  19. Translate(B, u) → B'
  20. Translate(C, u) → C'
  21. Translate(D, u) → D'
  22. Polygon(A', B', C', D') → poly1', a', b', c', d'
  23. Translate(A, v) → A'_1
  24. Translate(B, v) → B'_1
  25. Translate(C, v) → C'_1
  26. Translate(D, v) → D'_1
  27. Polygon(A'_1, B'_1, C'_1, D'_1) → poly1'_1, a'_1, b'_1, c'_1, d'_1
  28. Translate(A, w) → A'_2
  29. Translate(B, w) → B'_2
  30. Translate(C, w) → C'_2
  31. Translate(D, w) → D'_2
  32. Polygon(A'_2, B'_2, C'_2, D'_2) → poly1'_2, a'_2, b'_2, c'_2, d'_2
  33. Translate(A, z) → A'_3
  34. Translate(B, z) → B'_3
  35. Translate(C, z) → C'_3
  36. Translate(D, z) → D'_3
  37. Polygon(A'_3, B'_3, C'_3, D'_3) → poly1'_3, a'_3, b'_3, c'_3, d'_3
  38. Translate(A, e) → A'_4
  39. Polygon(O, B'_4, C'_4, D'_4) → poly1'_4, j_2, m_2, q_2, r_2
  40. Point(c') → J
  41. Line(J, a') → f
  42. Line(J, E_2) → g
  43. Line(B', g) → h
  44. Intersect(f, h) → K
  45. Midpoint(C', E_2) → L
  46. Midpoint(K, B') → N
  47. Midpoint(J, E_2) → M
  48. Segment(M, N) → i
  49. Point(i) → O_2
  50. Circle(O_2, Distance[N, K]) → q
  51. Line(O_2, h) → j
  52. Intersect(q, j) → P, Q
  53. Line(Q, d') → l_2
  54. Intersect(f, l_2) → R_2
  55. Polygon(R_2, P, Q) → poly3, q_1, r, p_1
  56. Midpoint(F_2, B'_1) → S
  57. Midpoint(c'_1) → T
  58. Midpoint(a'_1) → U
  59. Point(c'_2) → V
  60. Line(V, d'_2) → m
  61. Intersect(m, a'_2) → W
  62. Mirror(C, W) → C'_5
  63. Mirror(W, V) → W'
  64. Mirror(S, T) → S'
  65. Polygon(W', G, B'_2) → poly4, b'_6, w', g_1
  66. Polygon(S', D'_1, S, C'_1) → poly5, s', d'_5, s, c'_6
  67. Intersect(g_1, c'_2) → Z
  68. Intersect(c'_2, b'_6) → A_1
  69. Polygon(Z, C'_2, B'_2) → poly6, b'_7, z_1, c'_7
  70. Polygon(D'_2, A_1, G) → poly7, g_2, d'_6, a_2
  71. Line(D'_3, C'_3) → n_2
  72. Point(n_2) → B_1
  73. Point(n_2) → C_1
  74. Polygon(B_1, H, B'_3, C_1) → poly8, b_2, h_1, b'_8, c_2
  75. Polygon(A_1, Z, B'_2, G) → poly9, a_3, z_2, b'_9, g_4
  76. Polygon(J, K, B', E_2) → poly10, j_1, k_1, b'_{10}, e_2
  77. Segment(S', S) → p
  78. Point(p) → H_1
  79. Midpoint(c'_6) → I_1
  80. Midpoint(d'_5) → J_1
  81. Midpoint(s') → K_1
  82. Midpoint(s) → L_1
  83. Vector(I_1, J_1) → t
  84. Vector(K_1, L_1) → u_1
  85. Polygon(D'_1, T, S') → poly11, s'_1, d'_7, t_1
  86. Translate(D'_1, Vector[(Distance[S', H_1] / Distance[S', S] * u_1)]) → M_1
  87. Translate(T, Vector[(Distance[S', H_1] / Distance[S', S] * u_1)]) → T'
  88. Translate(S', Vector[(Distance[S', H_1] / Distance[S', S] * u_1)]) → S''
  89. Polygon(M_1, T', S'') → poly11', s'', m_1, t'
  90. Polygon(S', T, C'_1) → poly12, c'_8, s'_2, t_2
  91. Translate(S', Vector[(Distance[S', H_1] / Distance[S', S] * t)]) → S''_1
  92. Translate(T, Vector[(Distance[S', H_1] / Distance[S', S] * t)]) → T'_1
  93. Translate(C'_1, Vector[(Distance[S', H_1] / Distance[S', S] * t)]) → N_1
  94. Polygon(S''_1, T'_1, N_1) → poly12', n_1, s''_1, t'_1
  95. Polygon(D'_1, C'_1, S) → poly13, s_1, d'_8, c'_9
  96. Segment(V, W) → i_1
  97. Semicircle(W', B'_2) → r_1
  98. Semicircle(G, W') → f_2
  99. Point(r_1) → O_1
  100. Segment(O_1, Z) → l_1
  101. Semicircle(V, C'_2) → h_2
  102. Angle(B'_2, Z, C'_2) → α
  103. Rotate(O_1, α, Z) → D_1
  104. Segment(Z, D_1) → e_1
  105. Intersect(h_2, e_1) → E_1,
  106. Polygon(O_1, Z, E_1) → poly14, e_3, o_1, z_3
  107. Point(f_2) → F_1
  108. Semicircle(D'_2, V) → f_1
  109. Angle(D'_2, A_1, G) → β
  110. Rotate(F_1, (-β), A_1) → G_1
  111. Segment(G_1, A_1) → d_2
  112. Intersect(f_1, d_2) → P_1,
  113. Polygon(F_1, P_1, A_1) → poly15, a_4, f_3, p_2
  114. Polygon(A_1, V, W') → poly16, w'_1, a_5, v_1
  115. Midpoint(b'_8) → Q_1
  116. Mirror(H, B'_3) → H'
  117. Line(B_1, C_1) → i_2
  118. Mirror(H, Q_1) → H'_1
  119. CircumcircleArc(H, B_1, H'_1) → k_2
  120. Point(k_2) → R_1
  121. Angle(R_1, Q_1, H) → γ
  122. Rotate(B_1, (-γ), Q_1) → S_1
  123. Rotate(H, (-γ), Q_1) → H'_2
  124. Rotate(B'_3, (-γ), Q_1) → T_1
  125. Rotate(C_1, (-γ), Q_1) → U_1
  126. Polygon(S_1, H'_2, T_1, U_1) → poly8', s_2, h'_2, t_3, u_2
  127. Point(a) → F
  128. Sequence((x(O) + (R * sin((i * θ))), y(O) + R - (R * cos((i * θ)))), i, 0, (2 * n)) → list1
  129. Min((x(F) - x(E)) / (a / 2), 1) → k
  130. Sequence(If[k < 1, (x(O) + ((l_3 * sin((i * φ))) * cos((i * φ)) / sin(φ)), y(O) + ((l_3 * sin((i * φ))) * sin((i * φ)) / sin(φ))), (x(O) + (l_3 * i), y(O))], i, 0, (2 * n)) → list2
  131. Max((x(F) - x(E)) / (a / 2) - 1, 0) → k'
  132. Sequence(Sector[Rotate[Translate[Circle[(x(O) + 0, y(O) + R), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]]], i, 1, n, 1) → liste1
  133. Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O) + 0, y(O) + R), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l_3) - (R * sin(θ / 2) / 2), y(O) + (R * cos(θ / 2) / 2))], Rotate[Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l_3) - (R * sin(θ / 2) / 2), y(O) + (R * cos(θ / 2) / 2))], Rotate[Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l_3) - (R * sin(θ / 2) / 2), y(O) + (R * cos(θ / 2) / 2))]], i, n + 1, (2 * n)) → liste2
  134. Midpoint(m_2) → V_1
  135. Midpoint(r_2) → W_1
  136. Segment(V_1, W_1) → c_3
  137. Midpoint(C, A) → Z_1
  138. Midpoint(V_1, B'_4) → A_2
  139. Midpoint(j_2) → B_2
  140. Midpoint(J, B') → C_2
  141. Midpoint(a') → D_2
  142. Midpoint(p_1) → G_2
  143. Segment(H_1, S) → d_3
  144. Midpoint(b'_9) → H_2
  145. Midpoint(c_1) → I_2
  146. Midpoint(b_1) → J_2
  147. Midpoint(L_2, K_2) → M_2
  148. Vector(M_2, L_2) → l
  149. Vector(M_2, K_2) → w_1
  150. Midpoint(s_1) → N_2
  151. Vector(M_2, P_2) → v_2
  152. Midpoint(K_2, P_2) → Q_2
  153. Midpoint(S_2, T_2) → U_2
  154. Vector(U_2, T_2) → w_2
  155. Vector(U_2, S_2) → u_3
  156. OrthogonalLine(V, b'_9) → g_3
  157. Intersect(g_3, b'_9) → V_2
  158. Segment(V, V_2) → h_3
  159. Midpoint(h_3) → W_2
  160. OrthogonalLine(B_1, h_1) → i_3
  161. Intersect(i_3, h_1) → Z_2
  162. Segment(B_1, Z_2) → j_3
  163. Midpoint(j_3) → A_3
  164. Midpoint(B_1, C_1) → B_3
  165. Midpoint(h_1) → C_3
  166. Segment(W_1, O) → k_3
  167. Sequence(Segment[Element[list2, i], Element[list2, i + 1]], i, 1, Length[list2]) → liste3
  168. Segment(D_3, E_3) → m_3
  169. Segment(F_3, G_3) → n_3
  170. Segment(H_3, I_3) → p_3
  171. Midpoint(d_1) → J_3
  172. Midpoint(J_3, W_1) → K_3
  173. Midpoint(W_1, K_3) → L_3
  174. Midpoint(K_3, J_3) → M_3
  175. Midpoint(b_1) → N_3
  176. Midpoint(e_2) → O_3
  177. Midpoint(N_3, O_3) → P_3
  178. Midpoint(N_3, P_3) → Q_3
  179. Midpoint(P_3, O_3) → R_3
  180. Midpoint(N_3, D'_1) → S_3
  181. Midpoint(S_3, N_3) → T_3
  182. Midpoint(S_3, D'_1) → U_3
  183. Vector(Q_3, R_3) → v_3
  184. Vector(T_3, U_3) → w_3
  185. Vector(M_3, L_3) → q_3
  186. Midpoint(A, C_1) → V_3
  187. Midpoint(A, V_3) → W_3
  188. Midpoint(V_3, C_1) → Z_3
  189. Vector(W_3, Z_3) → r_3
  190. Midpoint(a_1) → A_4
  191. Midpoint(z_2) → B_4
  192. Midpoint(A_4, B_4) → C_4
  193. Midpoint(A_4, C_4) → D_4
  194. Midpoint(C_4, B_4) → E_4
  195. Vector(D_4, E_4) → s_3

AI Drawing Prompt Reference

Draw a quadrilateral determined by a point on the x-axis and a slanted line, then construct perpendiculars to form a complete shape. Translate this quadrilateral by several distinct vectors to create multiple congruent copies, each drawn as a polygon. On one translated copy, pick a point on a side, draw auxiliary lines through it and other vertices, find midpoints of certain segments, and connect those midpoints with a segment to illustrate area relationships under reflection and translation.
The prompt above was compiled from the original drawing commands and is for reference only.