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Section 3.3 Motions: Translating, Rotating, and Reflecting Lines, Circles, and P

Section 3.3 Motions: Translating, Rotating, and Reflecting Lines, Circles, and P
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2023-03-22 20:51
CirclePolygonVectorTransformationGeometric Construction
In collections geometriadinamica.es
This GeoGebra example demonstrates planar “motions” by visually transforming geometric configurations, showing how key points and entire polygons change under translation, rotation, and reflection (mirror). It centers on concepts of geometric transformation and construction: creating base objects with Line and Circle, locating critical positions via Intersect (line–circle or line–line intersections), and assembling shapes using Polygon. The construction uses Point and Vector to define and control moving reference elements (such as A, B, and grouped points like d/e/f/g and H/I), then applies Translate, Rotate, and Mirror to produce transformed counterparts (e.g., poly1 to poly1', poly2 to poly2'). Midpoint is used to support symmetry/center-related constructions. Suitable for secondary-school geometry lessons on transformations and for students learning how rigid motions preserve structure.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Circle(A, b) → c
  3. Intersect(c, a, 2) → C
  4. Polygon(A, C, 4) → poly1, d, e, f, g, D, E
  5. Point(f) → F
  6. Point(g) → G
  7. Point(d) → H
  8. Point(e) → I
  9. Polygon(H, I, F, G) → poly2, h, i, f_1, g_1
  10. Vector(E, J) → u
  11. Translate(A, u) → A'
  12. Translate(C, u) → C'
  13. Translate(D, u) → D'
  14. Translate(E, u) → E'
  15. Polygon(A', C', D', E') → poly1', a', c', d', e'
  16. Translate(H, u) → H'
  17. Translate(I, u) → I'
  18. Translate(F, u) → F'
  19. Translate(G, u) → G'
  20. Polygon(H', I', F', G') → poly2', h', i', f', g'
  21. Vector(D, K) → v
  22. Vector(L, M) → w
  23. Vector(L, N) → z
  24. Translate(A, v) → A'_1
  25. Translate(C, v) → C'_1
  26. Translate(D, v) → D'_1
  27. Translate(E, v) → E'_1
  28. Polygon(A'_1, C'_1, D'_1, E'_1) → poly1'_1, a'_1, c'_1, d'_1, e'_1
  29. Translate(H, v) → H'_1
  30. Translate(I, v) → I'_1
  31. Translate(F, v) → F'_1
  32. Translate(G, v) → G'_1
  33. Polygon(H'_1, I'_1, F'_1, G'_1) → poly2'_1, h'_1, i'_1, f'_1, g'_1
  34. Vector(E, O) → j
  35. Vector(D, P) → k
  36. Translate(A, j) → A'_2
  37. Translate(C, j) → C'_2
  38. Translate(D, j) → D'_2
  39. Translate(E, j) → E'_2
  40. Polygon(A'_2, C'_2, D'_2, E'_2) → poly1'_2, a'_2, c'_2, d'_2, e'_2
  41. Translate(H, j) → H'_2
  42. Translate(I, j) → I'_2
  43. Translate(F, j) → F'_2
  44. Translate(G, j) → G'_2
  45. Polygon(H'_2, I'_2, F'_2, G'_2) → poly2'_2, h'_2, i'_2, f'_2, g'_2
  46. Translate(A, k) → A'_3
  47. Translate(C, k) → C'_3
  48. Translate(D, k) → D'_3
  49. Translate(E, k) → E'_3
  50. Polygon(A'_3, C'_3, D'_3, E'_3) → poly1'_3, a'_3, c'_3, d'_3, e'_3
  51. Translate(H, k) → H'_3
  52. Translate(I, k) → I'_3
  53. Translate(F, k) → F'_3
  54. Translate(G, k) → G'_3
  55. Polygon(H'_3, I'_3, F'_3, G'_3) → poly2'_3, h'_3, i'_3, f'_3, g'_3
  56. Translate(A', w) → A''
  57. Translate(C', w) → C''
  58. Translate(D', w) → D''
  59. Translate(E', w) → E''
  60. Polygon(A'', C'', D'', E'') → poly1'', a'', c'', d'', e''
  61. Translate(A', z) → A''_1
  62. Translate(C', z) → C''_1
  63. Translate(D', z) → D''_1
  64. Translate(E', z) → E''_1
  65. Polygon(A''_1, C''_1, D''_1, E''_1) → poly1''_1, a''_1, c''_1, d''_1, e''_1
  66. Translate(A'', z) → A'''
  67. Translate(C'', z) → C'''
  68. Translate(D'', z) → D'''
  69. Translate(E'', z) → E'''
  70. Polygon(A''', C''', D''', E''') → poly1''', a''', c''', d''', e'''
  71. Translate(A'_1, w) → Q
  72. Translate(C'_1, w) → R
  73. Translate(D'_1, w) → S
  74. Translate(E'_1, w) → T
  75. Polygon(Q, R, S, T) → poly3, q, r, s, t
  76. Translate(A'_1, z) → U
  77. Translate(C'_1, z) → V
  78. Translate(D'_1, z) → W
  79. Translate(E'_1, z) → Z
  80. Polygon(U, V, W, Z) → poly4, u_1, v_1, w_1, z_1
  81. Translate(Q, z) → Q'
  82. Translate(R, z) → R'
  83. Translate(S, z) → S'
  84. Translate(T, z) → T'
  85. Polygon(Q', R', S', T') → poly3', q', r', s', t'
  86. Translate(A'_3, w) → A_1
  87. Translate(C'_3, w) → B_1
  88. Translate(D'_3, w) → C_1
  89. Translate(E'_3, w) → D_1
  90. Polygon(A_1, B_1, C_1, D_1) → poly5, a_1, b_1, c_1, d_1
  91. Translate(A'_3, z) → E_1
  92. Translate(C'_3, z) → F_1
  93. Translate(D'_3, z) → G_1
  94. Translate(E'_3, z) → H_1
  95. Polygon(E_1, F_1, G_1, H_1) → poly6, e_1, f_2, g_2, h_1
  96. Translate(A_1, z) → I_1
  97. Translate(B_1, z) → J_1
  98. Translate(C_1, z) → K_1
  99. Translate(D_1, z) → L_1
  100. Polygon(I_1, J_1, K_1, L_1) → poly5', i_1, j_1, k_1, l_1
  101. Translate(A'_2, w) → M_1
  102. Translate(C'_2, w) → N_1
  103. Translate(D'_2, w) → O_1
  104. Translate(E'_2, w) → P_1
  105. Polygon(M_1, N_1, O_1, P_1) → poly7, m_1, n_1, o_1, p_1
  106. Translate(A'_2, z) → Q_1
  107. Translate(C'_2, z) → R_1
  108. Translate(D'_2, z) → S_1
  109. Translate(E'_2, z) → T_1
  110. Polygon(Q_1, R_1, S_1, T_1) → poly8, q_1, r_1, s_1, t_1
  111. Translate(M_1, z) → U_1
  112. Translate(N_1, z) → V_1
  113. Translate(O_1, z) → W_1
  114. Translate(P_1, z) → Z_1
  115. Polygon(U_1, V_1, W_1, Z_1) → poly7', u_2, v_2, w_2, z_2
  116. Translate(H', w) → H''
  117. Translate(I', w) → I''
  118. Translate(F', w) → F''
  119. Translate(G', w) → G''
  120. Polygon(H'', I'', F'', G'') → poly2'', h'', i'', f'', g''
  121. Translate(H'', z) → H'''
  122. Translate(I'', z) → I'''
  123. Translate(F'', z) → F'''
  124. Translate(G'', z) → G'''
  125. Polygon(H''', I''', F''', G''') → poly2''', h''', i''', f''', g'''
  126. Translate(H', z) → H''_1
  127. Translate(I', z) → I''_1
  128. Translate(F', z) → F''_1
  129. Translate(G', z) → G''_1
  130. Polygon(H''_1, I''_1, F''_1, G''_1) → poly2''_1, h''_1, i''_1, f''_1, g''_1
  131. Mirror(H'_1, t) → A_2
  132. Mirror(I'_1, t) → B_2
  133. Mirror(F'_1, t) → C_2
  134. Mirror(G'_1, t) → D_2
  135. Polygon(A_2, B_2, C_2, D_2) → poly9, a_2, b_2, c_2, d_2
  136. Mirror(A_2, s') → E_2
  137. Mirror(B_2, s') → F_2
  138. Mirror(C_2, s') → G_2
  139. Mirror(D_2, s') → H_2
  140. Polygon(E_2, F_2, G_2, H_2) → poly9', e_2, f_3, g_3, h_2
  141. Mirror(H'_1, w_1) → I_2
  142. Mirror(I'_1, w_1) → J_2
  143. Mirror(F'_1, w_1) → K_2
  144. Mirror(G'_1, w_1) → L_2
  145. Polygon(I_2, J_2, K_2, L_2) → poly10, i_2, j_2, k_2, l_2
  146. Intersect(c'_3, d_1) → M_2
  147. Intersect(c'_3, d_1) → N_2
  148. Midpoint(c'_3) → O_2
  149. Midpoint(a_1) → P_2
  150. Mirror(H'_3, O_2) → R_2
  151. Mirror(I'_3, O_2) → S_2
  152. Mirror(F'_3, O_2) → T_2
  153. Mirror(G'_3, O_2) → U_2
  154. Polygon(R_2, S_2, T_2, U_2) → poly11, r_2, s_2, t_2, u_3
  155. Mirror(R_2, P_2) → V_2
  156. Mirror(S_2, P_2) → W_2
  157. Mirror(T_2, P_2) → Z_2
  158. Mirror(U_2, P_2) → A_3
  159. Polygon(V_2, W_2, Z_2, A_3) → poly11', v_3, w_3, z_3, a_3
  160. Mirror(V_2, Q_2) → B_3
  161. Mirror(W_2, Q_2) → C_3
  162. Mirror(Z_2, Q_2) → D_3
  163. Mirror(A_3, Q_2) → E_3
  164. Polygon(B_3, C_3, D_3, E_3) → poly11'', b_3, c_3, d_3, e_3
  165. Rotate(H'_2, 270°, C'_2) → F_3
  166. Rotate(I'_2, 270°, C'_2) → G_3
  167. Rotate(F'_2, 270°, C'_2) → H_3
  168. Rotate(G'_2, 270°, C'_2) → I_3
  169. Polygon(F_3, G_3, H_3, I_3) → poly12, f_4, g_4, h_3, i_3
  170. Rotate(F_3, 270°, C'_2) → J_3
  171. Rotate(G_3, 270°, C'_2) → K_3
  172. Rotate(H_3, 270°, C'_2) → L_3
  173. Rotate(I_3, 270°, C'_2) → M_3
  174. Polygon(J_3, K_3, L_3, M_3) → poly12', j_3, k_3, l_3, m_3
  175. Rotate(J_3, 270°, C'_2) → N_3
  176. Rotate(K_3, 270°, C'_2) → O_3
  177. Rotate(L_3, 270°, C'_2) → P_3
  178. Rotate(M_3, 270°, C'_2) → Q_3
  179. Polygon(N_3, O_3, P_3, Q_3) → poly12'', n_3, o_3, p_3, q_3

AI Drawing Prompt Reference

Draw two line segments and two circles, find their intersection points, and use these points to construct two quadrilaterals. Mark the key points of these quadrilaterals, define translations and rotations using vectors. Translate and rotate the quadrilaterals along the vectors, and draw the transformed shapes to demonstrate the effects of geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.