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Tangram Challenges: Constructing a Puzzle Shape via Rotation

Tangram Challenges: Constructing a Puzzle Shape via Rotation
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2021-03-10 20:52
TriangleCirclePolygonTransformationGeometric Construction
This example demonstrates how to build a tangram-style puzzle configuration in GeoGebra by constructing key points from circles and lines, then assembling the final picture using geometric transforms. It focuses on geometric construction workflows: creating circles and baseline lines, finding intersections to determine vertices, using orthogonal helper lines to fix triangle placement, and forming polygons that represent the assembled regions. The underlying math includes standard geometry tools (circle/line/segment/ray), intersection-based construction (Intersect), orthogonal construction (OrthogonalLine), polygon composition (Polygon), and especially rotation-driven transformation (Rotate) to generate multiple congruent triangle parts and align them into the target tangram challenge. The construction uses conditional logic (If) to manage object creation or visibility. Suitable for middle school through early university students, and ideal for instruction and exploration of geometry construction and transformation-based reasoning.

GeoGebra Commands & Parameters

  1. Circle(A, a) → c
  2. Point(c) → B
  3. Line(A, B) → f
  4. OrthogonalLine(B, f) → g_{4}
  5. Circle(B, A) → d
  6. Intersect(d, g_{4}, 2) → C
  7. Polygon(A, B, C) → t1, c_1, a_1, b_{3}
  8. Circle(D, a) → e
  9. Point(e) → E
  10. Line(D, E) → h
  11. OrthogonalLine(E, h) → i
  12. Circle(E, D) → k
  13. Intersect(k, i, 2) → F
  14. Polygon(D, E, F) → t2, f_1, d_1, e_1
  15. Circle(G, a / 2) → p
  16. Point(p) → H
  17. Line(G, H) → j
  18. OrthogonalLine(H, j) → l
  19. Circle(H, G) → q
  20. Intersect(q, l, 2) → I
  21. OrthogonalLine(G, j) → m
  22. OrthogonalLine(I, m) → n
  23. Intersect(m, n) → J
  24. Polygon(G, H, I, J) → q1, g_1, h_1, i_1, j_1
  25. Circle(N, a / 2) → p_1
  26. Point(p_1) → O
  27. Line(N, O) → n_1
  28. OrthogonalLine(O, n_1) → q_1
  29. Circle(O, N) → r_1
  30. Intersect(r_1, q_1, 1) → P
  31. Polygon(O, N, P) → t4, p_2, o_1, n_2
  32. Circle(Q, (sqrt(2) / 2 * a)) → s_1
  33. Point(s_1) → R
  34. Line(Q, R) → t_1
  35. Rotate(R, ((-π)) / 4, Q) → R'
  36. Rotate(R, π / 4, Q) → R'_{1}
  37. Ray(Q, R') → b_1
  38. Circle(Q, a / 2) → d_2
  39. Intersect(d_2, b_1, 1) → S
  40. Line(S, t_1) → f_2
  41. Circle(S, (sqrt(2) / 2 * a)) → c_2
  42. Intersect(c_2, f_2, 2) → T
  43. Polygon(Q, S, T, R) → q2, q_2, s_2, t_2, r_2
  44. Ray(Q, R'_{1}) → g_2
  45. Intersect(d_2, g_2, 1) → U
  46. Line(U, t_1) → h_2
  47. Circle(U, (sqrt(2) / 2 * a)) → e_2
  48. Intersect(e_2, h_2, 2) → V
  49. Polygon(Q, R, V, U) → q3, q_3, r_3, v, u
  50. Point(Circle[W, a / 2]) → Z
  51. Segment(W, Z) → i_2
  52. Rotate(Z, ((-π)) / 2, W) → Z'
  53. Polygon(Z, W, Z') → t5, z', z_1, w
  54. Point(Circle[A_1, a]) → B_1
  55. Segment(A_1, B_1) → j_2
  56. Rotate(B_1, ((-π)) / 4, A_1) → C_1
  57. Rotate(A_1, π / 4, B_1) → D_1
  58. Ray(B_1, D_1) → m_2
  59. Ray(A_1, C_1) → a_2
  60. Intersect(m_2, a_2) → E_1
  61. Polygon(B_1, A_1, E_1) → t6, b_2, f_3, g_3
  62. If(o ≟ 1, (sqrt(2) / 32 * a), o ≟ 2, a / 32) → r_{4}
  63. Point(Circle[K, (a * sqrt(2))]) → L
  64. Segment(K, L) → k_1
  65. Polygon(K, L, 4) → poly1, l_1, m_1, k_2, l_2, M, F_1

AI Drawing Prompt Reference

Draw a tangram-style puzzle shape. Start by creating a circle with a given radius, pick a point on it, draw a line from the center to that point, then construct a perpendicular line through that point; the intersection with the circle gives a third vertex, forming an isosceles right triangle. Repeat this process elsewhere to make a second congruent triangle. Next, construct a square with side length half the radius, using perpendicular lines and intersections to define its four vertices. Then, create a smaller isosceles right triangle, and a quadrilateral formed by rotating a radius endpoint by 45 degrees, drawing rays to intersect a circle, and connecting the resulting points. Finally, assemble all polygons into the complete tangram configuration.
The prompt above was compiled from the original drawing commands and is for reference only.