MathPlotAI Logo MathPlotAI

GoGeometry 191: Comprehensive Geometric Transformations

GoGeometry 191: Comprehensive Geometric Transformations
Loading the math board and drawing, please wait…
2021-05-03 10:05
TrianglePolygonVectorSolid GeometryGeometric Construction
This example demonstrates how to use GeoGebra to construct a comprehensive geometric figure including parallelograms, squares, isosceles triangles, rhombuses, and 3D vectors. Core mathematical concepts cover plane geometry properties, vector operations, geometric transformations (rotation, dilation, translation, reflection), and 3D vector representation. Construction key points: Use Point for key vertices, Polygon for shapes, Dilate for scaling, Rotate for rotation, Vector and Translate for vector operations, Mirror for reflection, CircleArc and CircleSector for arcs and sectors, Midpoint for midpoint finding, and OrthogonalLine for perpendicular lines. This example is suitable for middle school geometry teaching, vector concept instruction, and GeoGebra skill training, demonstrating relationships and transformation rules between geometric figures.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(Ray[A, A + (1, 0)]) → B_2
  3. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  4. Segment(B_2, C) → h
  5. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  6. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  7. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  8. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  9. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  10. Segment(R, A''') → f''
  11. Segment(A, B_2) → g
  12. Circle(B_2, g) → q_2
  13. Ray(A, B_2) → m_2
  14. Intersect(q_2, m_2, 2) → J_1
  15. Segment(B_2, J_1) → n_2
  16. Point(Ray[B_2, B_2 + (1, 0)]) → K_1
  17. Segment(B_2, K_1) → r_2
  18. Circle(K_1, r_2) → t_2
  19. OrthogonalLine(K_1, r_2) → s_2
  20. Intersect(t_2, s_2, 2) → L_1
  21. OrthogonalLine(L_1, h) → f_3
  22. Intersect(f_3, h) → M_1
  23. Dilate(A, If[17 < e ≤ 18, 18 - e, 0], M_1) → A'_4
  24. Line(K_1, M_1) → i_2
  25. Midpoint(A, C) → V
  26. CircleArc(V, C, D) → g_2
  27. Point(g_2) → B
  28. Segment(C, B) → j_1
  29. Segment(A, B) → n
  30. Intersect(n, f'') → Z
  31. Segment(B, Z) → l_1
  32. Segment(A, A') → f
  33. Segment(A'_1, A'') → f'
  34. Segment(C, D) → i
  35. Circle(B_2, (0.25 * r_2)) → p
  36. Segment(D, A) → j
  37. Circle(C, Radius[p]) → k
  38. Circle(A, (0.1 * j)) → q
  39. Circle(D, (0.1 * j)) → r
  40. Segment(A, C) → l
  41. Segment(D, B_2) → m
  42. Intersect(r, j, 1) → F
  43. Intersect(r, m, 1) → G
  44. Intersect(r, i, 1) → H
  45. Intersect(k, i, 1) → I
  46. Intersect(k, l, 1) → J
  47. Intersect(k, h, 1) → K
  48. Intersect(p, h, 1) → L
  49. Intersect(p, m, 1) → M
  50. Intersect(p, g, 1) → N
  51. Intersect(q, g, 1) → O
  52. Intersect(q, l, 1) → P
  53. Intersect(q, j, 1) → Q
  54. CircleSector(B_2, L, N) → s
  55. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  56. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  57. Segment(R', S) → f'''
  58. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  59. Vector(B_2, B') → u
  60. Translate(s, u) → s'
  61. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  62. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  63. Vector(C, C') → v
  64. Translate(s'', v) → s'''
  65. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  66. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  67. Vector(D, D') → w
  68. Translate(t, w) → t'
  69. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  70. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  71. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  72. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  73. Vector(B_2, B'_1) → v_1
  74. Vector(B_2, B'_3) → w_1
  75. Vector(B_2, B'_2) → i_1
  76. Circle(A, n - j_1) → h_2
  77. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  78. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  79. Vector(B_2, B'_4) → u_1
  80. Translate(s, u_1) → s'_1
  81. Line(B, f') → a_1
  82. Point(a_1) → U
  83. Angle(U, B, C) → α
  84. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  85. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  86. Line(A, B_2) → h_1
  87. Angle(B, A, D) → β
  88. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  89. Mirror(C, D) → C'_1
  90. Rotate(C, ((-π)) / 3, B_2) → C'_2
  91. Segment(B_2, C'_2) → f_1
  92. Segment(C, C'_2) → g_1
  93. Intersect(k, g_1, 1) → E
  94. Intersect(k, h) → T,
  95. CircleSector(C, T, E) → d_1
  96. Rotate(D, 15°, C) → D'_2
  97. Angle(D, C, D'_2) → γ
  98. Ray(C, D'_2) → k_1
  99. Rotate(C, (-(15°)), D) → C'_3
  100. Angle(C'_3, D, C) → δ
  101. Ray(D, C'_3) → m_1
  102. Intersect(k_1, m_1) → W
  103. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  104. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  105. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  106. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  107. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  108. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  109. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  110. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  111. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  112. Rotate(T, 15°, C) → T'
  113. Angle(T, C, T') → ε
  114. Ray(C, T') → q_1
  115. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  116. Vector(C, C'_8) → u_2
  117. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  118. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  119. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  120. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  121. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  122. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  123. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  124. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  125. Segment(B_2, C_1) → n_1
  126. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  127. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  128. Segment(D_1, E_1) → p_1
  129. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  130. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  131. Segment(F_1, N_1) → r_1
  132. Rotate(F_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → O_1
  133. Rotate(N_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → P_1
  134. Segment(O_1, P_1) → s_1
  135. Polygon(B_2, K_1, L_1, M_1) → poly6, b_3, k_2, l_3, m_3
  136. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  137. Vector(B_2, Q_1) → c_1
  138. Translate(s, c_1) → s'_2
  139. Segment(C, K_1) → e_1
  140. Line(A, M_1) → t_1
  141. Intersect(t_1, e_1) → R_1
  142. Segment(A, R_1) → f_2
  143. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  144. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], M_1) → A'_3
  145. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], R_1) → S_1
  146. Line(R_1, p_1) → g_3
  147. Point(g_3) → U_1
  148. Angle(U_1, R_1, K_1) → ζ
  149. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4
  150. Dilate(K_1, If[15 < e ≤ 16, 16 - e, 0], M_1) → W_1
  151. Line(A, K_1) → k_3
  152. Rotate(D_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → H_2
  153. Rotate(E_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → I_2
  154. Segment(H_2, I_2) → f_4
  155. Rotate(B_2, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → J_2
  156. Rotate(K_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → K_2
  157. Rotate(L_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → L_2
  158. Rotate(M_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → M_2
  159. Polygon(J_2, K_2, L_2, M_2) → poly6'_{1}, g_4, h_4, i_4, j_4
  160. Rotate(F_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → N_2
  161. Rotate(N_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → O_2
  162. Segment(N_2, O_2) → k_4
  163. Rotate(O_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → P_2
  164. Rotate(P_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → Q_2
  165. Segment(P_2, Q_2) → l_4
  166. Rotate(B_2, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → R_2
  167. Rotate(C_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → S_2
  168. Segment(R_2, S_2) → m_4
  169. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], D_1) → A'_{7}
  170. Segment(A, A'_{7}) → p_2
  171. Rotate(A, If[17 < e ≤ 18, (((-π)) / 2 * (e - 17)), ((-π)) / 2], D_1) → A'_{6}
  172. Segment(D_1, A'_{6}) → e_2
  173. Dilate(D_1, If[18 < e < -19, 19 - e, 0], A'_{6}) → T_1
  174. Dilate(D_1, If[18 < e < 19, 19 - e, 0], A'_{6}) → V_1
  175. Vector(D_1, V_1) → v_2
  176. Translate(A, v_2) → A'_{5}
  177. Translate(A'_{7}, v_2) → Z_1
  178. Segment(A'_{5}, Z_1) → j_3
  179. Dilate(D_1, If[19 < e ≤ 20, 20 - e, 0], A) → A_2
  180. Vector(D_1, A_2) → u_3
  181. Translate(D_1, u_3) → C_2
  182. Translate(A'_{6}, u_3) → D_2
  183. Segment(C_2, D_2) → n_3
  184. Dilate(R, If[20 < e < 21, 21 - e, 0], T_1) → R'_{1}
  185. Vector(R, R'_{1}) → v_3
  186. Translate(s'', v_3) → s'''_{1}
  187. Line(T_1, f'') → p_3
  188. Point(p_3) → E_2
  189. Angle(E_2, T_1, A'_{5}) → η
  190. Rotate(s'''_{1}, If[20 < e ≤ 21, ((-(e - 20)) * ((2 * π) - η)), (-((2 * π) - η))], R'_{1}) → e_3
  191. Dilate(A'_{5}, If[21 < e ≤ 22, 22 - e, 0], D) → F_2
  192. Vector(A'_{5}, F_2) → q_3
  193. Segment(A'_{5}, F_2) → r_3
  194. Dilate(D, If[22 < e ≤ 23, 23 - e, 0], H_2) → D'_{4}
  195. Segment(D, D'_{4}) → s_3
  196. Dilate(H_2, If[23 < e ≤ 24, 24 - e, 0], D_1) → G_2
  197. Vector(H_2, G_2) → t_3
  198. Translate(D, t_3) → D'_{5}
  199. Translate(D'_{4}, t_3) → T_2
  200. Segment(D'_{5}, T_2) → n_4
  201. Dilate(A'_{5}, If[24 < e ≤ 25, 25 - e, 0], D_1) → U_2
  202. Vector(A'_{5}, U_2) → u_4
  203. Translate(A'_{5}, u_4) → V_2
  204. Translate(F_2, u_4) → W_2
  205. Segment(V_2, W_2) → p_4
  206. Polygon(A'_{5}, D, H_2, D_1) → q1, a'_{5}, d_5, j_2, l_2

AI Drawing Prompt Reference

Construct a comprehensive geometric figure: start with a square, then create a dynamic triangle on one side that undergoes scaling, rotation, and reflection based on a parameter. Outside the square, add a circle and a perpendicular line to form intersection points, and connect relevant segments. Draw an arc centered at the midpoint of the square's diagonal, pick a point on the arc, and connect it to the square's vertices to form a triangle. Finally, add several small circles near the square's sides with their intersection points to complete the transformation demonstration.
The prompt above was compiled from the original drawing commands and is for reference only.