MathPlotAI Logo MathPlotAI

Animation 109: Rotational Construction of Quadrilaterals and Isosceles Triangles

Animation 109: Rotational Construction of Quadrilaterals and Isosceles Triangles
Loading the math board and drawing, please wait…
2016-09-07 03:11
PolygonTrigonometryTransformationGeometric ConstructionSlider Animation
In collections CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2
This example demonstrates a dynamic geometric figure composed of squares, quadrilaterals, and isosceles triangles, generated through rotations, translations, and dilations to create interconnected shapes. Core mathematical concepts include polygons (squares, quadrilaterals, isosceles triangles), trigonometry (angle parameters), rotation and translation transformations, midpoints and intersections, and integer-parameter-driven dynamics. Construction highlights: from the summary, GeoGebra commands such as Polygon, Angle, Segment, Midpoint, Rotate, Min, Circle, Intersect, Dilate, CircleSector, Vector, Translate, Line, and Point are used to create multiple polygons (poly1 to poly5), segments, circles, sectors, vectors, and points, with angles α, β, γ and integer parameters driving the animation. Suitable for middle/high school geometry teaching or math enthusiasts exploring dynamic geometric transformations, useful for understanding rotational symmetry, isosceles triangle properties, and parametric construction.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Polygon(D, C, 4) → poly2, f, g, h, i, E, F
  3. Polygon(C, B, 4) → poly3, j, k, l, m, G, H
  4. Polygon(B, A, 4) → poly4, n, p, q, r, I, J
  5. Polygon(A, D, 4) → poly5, s, t, e, f_1, K, L
  6. Angle(poly2) → α, β, γ, δ
  7. Angle(poly5) → ε, ζ, η, θ
  8. Angle(poly4) → ι, κ, λ, μ
  9. Angle(poly3) → ν, ξ, ο, ρ
  10. Segment(A, D) → g_1
  11. Segment(D, C) → h_1
  12. Segment(C, B) → i_1
  13. Segment(B, A) → j_1
  14. Midpoint(C, F) → M
  15. Midpoint(D, L) → N
  16. Midpoint(A, J) → O
  17. Midpoint(B, H) → P
  18. Segment(O, M) → k_1
  19. Segment(P, N) → l_1
  20. Segment(D, F) → m_1
  21. Segment(F, E) → n_1
  22. Segment(E, C) → p_1
  23. Segment(C, H) → q_1
  24. Segment(H, G) → r_1
  25. Segment(G, B) → s_1
  26. Segment(A, I) → t_1
  27. Segment(A, L) → a_1
  28. Segment(L, K) → b_1
  29. Segment(K, D) → c_1
  30. Segment(J, I) → d_1
  31. Segment(J, B) → e_1
  32. Rotate(D, If[0 < o ≤ 1, (o * π / 2), π / 2], D) → D'
  33. Rotate(C, If[0 < o ≤ 1, (o * π / 2), π / 2], D) → C'
  34. Segment(D, F) → g_2
  35. Min({g_1, j_1, i_1, h_1}) → u
  36. Circle(D, (0.25 * u)) → c_2
  37. Intersect(c_2, h_1, 1) → Q
  38. Intersect(i, g_2) → R
  39. Intersect(i, g_2) → S
  40. Intersect(c_2, g_2) → T,
  41. Intersect(k_1, l_1) → U
  42. Circle(U, (0.2 * u)) → e_2
  43. Intersect(e_2, l_1, 1) → V
  44. Intersect(e_2, k_1, 2) → W
  45. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  46. Segment(A, A') → i_2
  47. Dilate(B, If[1 < o ≤ 2, 2 - o, 0], C) → B'
  48. Segment(B, B') → j_2
  49. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], D) → C'_1
  50. Segment(C, C'_1) → k_2
  51. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], A) → D'_1
  52. Segment(D, D'_1) → l_2
  53. Polygon(A, B, C, D) → poly6, a_2, b_2, c_3, d_3
  54. Circle(C, (0.2 * u)) → d_2
  55. Segment(C, E) → f_2
  56. Segment(C, D) → m_2
  57. Intersect(d_2, m_2) → Z,
  58. Intersect(d_2, f_2) → A_1,
  59. CircleSector(C, A_1, Z) → p_2
  60. Rotate(C, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], C) → C'_2
  61. Rotate(C'_1, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], C) → B_1
  62. Segment(C'_2, B_1) → n_2
  63. Rotate(B, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], B) → B'_1
  64. Rotate(B', If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], B) → B''
  65. Segment(B'_1, B'') → q_2
  66. Rotate(A, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], A) → A'_1
  67. Rotate(A', If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], A) → A''
  68. Segment(A'_1, A'') → r_2
  69. Rotate(D, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], D) → D'_2
  70. Rotate(D'_1, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], D) → C_1
  71. Segment(D'_2, C_1) → s_2
  72. Rotate(C'_2, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], E) → D_1
  73. Rotate(B_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], E) → E_1
  74. Segment(D_1, E_1) → t_2
  75. Rotate(B'_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], G) → F_1
  76. Rotate(B'', If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], G) → B'''
  77. Segment(F_1, B''') → f_3
  78. Rotate(A'_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], I) → G_1
  79. Rotate(A'', If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], I) → A'''
  80. Segment(G_1, A''') → g_3
  81. Rotate(D'_2, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], K) → H_1
  82. Rotate(C_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], K) → I_1
  83. Segment(H_1, I_1) → h_3
  84. Rotate(D_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], F) → J_1
  85. Rotate(E_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], F) → K_1
  86. Segment(J_1, K_1) → i_3
  87. Rotate(F_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], H) → L_1
  88. Rotate(B''', If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], H) → M_1
  89. Segment(L_1, M_1) → j_3
  90. Rotate(G_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], J) → N_1
  91. Rotate(A''', If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], J) → O_1
  92. Segment(N_1, O_1) → k_3
  93. Rotate(H_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], L) → P_1
  94. Rotate(I_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], L) → Q_1
  95. Segment(P_1, Q_1) → l_3
  96. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], E) → C'_3
  97. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], D) → C'_4
  98. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], A) → C'_5
  99. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], B) → C'_6
  100. Vector(C, C'_3) → v
  101. Vector(C, C'_4) → w
  102. Vector(C, C'_5) → u_1
  103. Vector(C, C'_6) → v_1
  104. Translate(p_2, v) → e_3
  105. Translate(p_2, w) → p_3
  106. Translate(p_2, u_1) → q_3
  107. Translate(p_2, v_1) → r_3
  108. Rotate(e_3, If[7 < o ≤ 8, ((o - 7) * π / 2), π / 2], C'_3) → s_3
  109. Angle(poly1) → σ, τ, υ, ϕ
  110. Line(D, C) → m_3
  111. Line(C, B) → n_3
  112. Line(B, A) → t_3
  113. Rotate(p_3, If[7 < o ≤ 8, ((-(o - 7)) * (π + ϕ)), (-(π + ϕ))], C'_4) → c_4
  114. Angle(H, C, E) → χ
  115. Rotate(r_3, If[7 < o ≤ 8, ((-(o - 7)) * χ), (-χ)], C'_6) → d_4
  116. Line(A, n_2) → a_3
  117. Point(a_3) → R_1
  118. Angle(R_1, A, I) → ψ
  119. Rotate(q_3, If[7 < o ≤ 8, ((o - 7) * ψ), ψ], C'_5) → e_4
  120. Dilate(E, If[8 < o ≤ 9, 9 - o, 0], F) → E'
  121. Dilate(B, If[8 < o ≤ 9, 9 - o, 0], G) → B'_2
  122. Dilate(A, If[8 < o ≤ 9, 9 - o, 0], I) → A'_2
  123. Dilate(D, If[8 < o ≤ 9, 9 - o, 0], K) → D'_3
  124. Vector(E, E') → w_1
  125. Vector(D, D'_3) → u_2
  126. Vector(A, A'_2) → v_2
  127. Vector(B, B'_2) → w_2
  128. Translate(s_3, w_1) → f_4
  129. Translate(c_4, u_2) → g_4
  130. Translate(e_4, v_2) → h_4
  131. Translate(d_4, w_2) → k_4
  132. Rotate(f_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], E') → p_4
  133. Rotate(g_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], D'_3) → q_4
  134. Rotate(h_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], A'_2) → r_4
  135. Rotate(k_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], B'_2) → s_4
  136. Dilate(F, If[9 < o ≤ 10, 10 - o, 0], D) → F'
  137. Dilate(K, If[9 < o ≤ 10, 10 - o, 0], L) → K'
  138. Dilate(I, If[9 < o ≤ 10, 10 - o, 0], J) → I'
  139. Dilate(G, If[9 < o ≤ 10, 10 - o, 0], H) → G'
  140. Vector(K, K') → u_3
  141. Vector(I, I') → v_3
  142. Vector(G, G') → w_3
  143. Vector(F, F') → b_3
  144. Translate(p_4, b_3) → t_4
  145. Translate(q_4, u_3) → c_5
  146. Translate(s_4, w_3) → d_5
  147. Translate(r_4, v_3) → e_5
  148. Rotate(t_4, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], F') → f_5
  149. Rotate(c_5, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], K') → g_5
  150. Rotate(e_5, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], I') → h_5
  151. Rotate(d_5, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], G') → k_5
  152. Rotate(p_2, If[10 < o ≤ 11, ((o - 10) * (π / 2 + υ)), π / 2 + υ], C) → p_5
  153. Rotate(e_4, If[10 < o ≤ 11, ((o - 10) * (π / 2 + σ)), π / 2 + σ], A) → q_5
  154. Rotate(d_4, If[10 < o ≤ 11, ((o - 10) * (π / 2 + τ)), π / 2 + τ], B) → r_5
  155. Dilate(C, If[11 < o ≤ 12, 12 - o, 0], M) → C'_7
  156. Dilate(B, If[11 < o ≤ 12, 12 - o, 0], P) → B'_3
  157. Dilate(A, If[11 < o ≤ 12, 12 - o, 0], O) → A'_3
  158. Dilate(D, If[11 < o ≤ 12, 12 - o, 0], N) → D'_4
  159. Segment(C, C'_7) → i_4
  160. Segment(D, D'_4) → j_4
  161. Segment(A, A'_3) → l_4
  162. Segment(B, B'_3) → m_4
  163. Rotate(C, If[12 < o ≤ 13, ((o - 12) * π), π], M) → C'_8
  164. Rotate(C'_7, If[12 < o ≤ 13, ((o - 12) * π), π], M) → S_1
  165. Segment(C'_8, S_1) → n_4
  166. Rotate(B, If[12 < o ≤ 13, ((o - 12) * π), π], P) → B'_4
  167. Rotate(B'_3, If[12 < o ≤ 13, ((o - 12) * π), π], P) → T_1
  168. Segment(B'_4, T_1) → a_4
  169. Rotate(A, If[12 < o ≤ 13, ((o - 12) * π), π], O) → A'_4
  170. Rotate(A'_3, If[12 < o ≤ 13, ((o - 12) * π), π], O) → U_1
  171. Segment(A'_4, U_1) → b_4
  172. Rotate(D, If[12 < o ≤ 13, ((o - 12) * π), π], N) → D'_5
  173. Rotate(D'_4, If[12 < o ≤ 13, ((o - 12) * π), π], N) → V_1
  174. Segment(D'_5, V_1) → i_5
  175. Dilate(M, If[13 < o ≤ 14, 14 - o, 0], O) → M'
  176. Segment(M, M') → h_2
  177. Rotate(M, If[14 < o ≤ 15, ((-(o - 14)) * π / 2), ((-π)) / 2], O) → M'_1
  178. Rotate(M', If[14 < o ≤ 15, ((-(o - 14)) * π / 2), ((-π)) / 2], O) → M''
  179. Segment(M'_1, M'') → j_5
  180. Dilate(O, If[15 < o ≤ 16, 16 - o, 0], N) → O'
  181. Vector(O, O') → u_4
  182. Translate(M'_1, u_4) → W_1
  183. Translate(M'', u_4) → M'''
  184. Segment(W_1, M''') → l_5
  185. Dilate(I, If[16 < o ≤ 17, 17 - o, 0], U) → I'_1
  186. Vector(I, I'_1) → v_4
  187. Translate(r_4, v_4) → s_5
  188. Line(I, h_2) → m_5
  189. Line(I, l_5) → n_5
  190. Point(m_5) → Z_1
  191. Angle(A, I, Z_1) → ω
  192. Rotate(s_5, If[16 < o ≤ 17, ((o - 16) * ω), ω], I'_1) → t_5
  193. Rotate(t_5, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], U) → c_6