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Axial Symmetry

Axial Symmetry
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2020-04-09 15:36
FunctionTransformationGeometric Construction
This example demonstrates the construction of axial symmetry figures. It involves key mathematical concepts such as axial symmetry and geometric construction. Using GeoGebra commands like PolyLine, Mirror, and Segment, axial symmetry figures can be constructed. Suitable for middle school and above students to learn about axial symmetry properties and geometric construction techniques.

GeoGebra Commands & Parameters

  1. PolyLine(O_1, P_1, A_2, O_1, Q_1, R_1, S_1, T_1, U_1, V_1, W_1, Z_1) → q_1
  2. Point(q_1) → P_3
  3. PolyLine(I, H, N, M, L, K, J, I, G, F, E, D, C, B, A, O) → t
  4. Point(t) → P1
  5. Polygon(W, Z, C_1, D_1, E_1, F_1, A_1, B_1, V) → poly2, w, z, c_1, d_1, e_1, f_1, a_1, b_1, v
  6. Point(v) → G_1
  7. Point(v) → H_1
  8. PolyLine(A_1, F_1, B_1, A_1, E_1, D_1, C_1, Z, W, H_1, G_1, V) → p_1
  9. Point(p_1) → P2
  10. PolyLine(B_2, C_2, L_2, M_2, B_2, D_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2) → r_1
  11. Point(r_1) → P4
  12. Polygon(A, B, K, L, M, N, O, C, D, E, F, G, H, I, J) → poly1, a_2, j, k, l, m, n, o, b, c, d, e, f, g, h, i
  13. Line(Q, R) → p
  14. If(Motif ≟ 1, P1, If[Motif ≟ 2, P2, If[Motif ≟ 3, P_3, If[Motif ≟ 4, P4, P5]]]) → P
  15. Mirror(P, p) → S
  16. Circle(P, a) → q
  17. Line(P, xAxis) → r
  18. Intersect(q, r) → T, U
  19. Sequence(Segment[Element[Marque, i], Element[Marque, i + 1]], i, 1, Length[Marque]) → Trait
  20. Mirror(Trait, p) → Trait'
  21. Polygon(H_1, I_1, J_1, G_1) → poly3, h_1, i_1, j_1, g_1
  22. Mirror(H_1, p) → K_1
  23. Mirror(I_1, p) → L_1
  24. Mirror(J_1, p) → M_1
  25. Mirror(G_1, p) → N_1
  26. Polygon(K_1, L_1, M_1, N_1) → poly3', k_1, l_1, m_1, n_1
  27. Segment(P, S) → s
  28. Intersect(s, p) → N_2
  29. Angle(S, N_2, R) → α
  30. Circle(N_2, 1) → s_1
  31. Intersect(s_1, p) → O_2, P_2
  32. Intersect(s_1, s) → Q_2, R_2
  33. Line(R_2, p) → t_1
  34. Line(O_2, s) → b_2
  35. Intersect(b_2, t_1) → S_2
  36. Polygon(N_2, R_2, S_2, O_2) → poly4, n_2, r_2, s_2, o_2
  37. Segment(N_2, P) → g_2
  38. Segment(N_2, S) → f_2
  39. Rotate(Trait', rr, p) → Trait''
  40. Midpoint(Q, R) → T_2
  41. Plane(U_2, Q, R) → p_2
  42. Point(p) → V_2
  43. Point(p) → W_2
  44. OrthogonalLine(V_2, p_2) → h_2
  45. OrthogonalLine(W_2, p_2) → i_2
  46. Point(h_2) → Z_2
  47. Line(Z_2, p) → j_2
  48. Mirror(Z_2, p) → A_3
  49. Intersect(i_2, j_2) → B_3
  50. Mirror(B_3, p) → C_3
  51. Polygon(V_2, A_3, C_3, W_2) → q1, v_2, a_3, c_3, w_2
  52. Polygon(V_2, Z_2, B_3, W_2) → q2, k_2, z_2, b_3, l_2
  53. Rotate(q2, rr, p) → q2'
  54. Rotate(S, rr, p) → S'
  55. Rotate(N_2, rr, p) → D_3
  56. Rotate(S, rr, p) → S'_1
  57. Segment(D_3, S'_1) → m_2

AI Drawing Prompt Reference

Draw an axial symmetry demonstration. Start with a straight line as the axis of symmetry. Construct two polygons on opposite sides of the axis, one open polyline and one closed polygon. Place a movable point on the open polyline, then draw a perpendicular from this point to the axis. Use a circle centered at the intersection of the perpendicular and the axis to construct a small square, illustrating the local symmetry operation. Mirror the edges of the closed polygon across the axis to create its reflected counterpart, and connect corresponding points with dashed segments. Mark the angle between the perpendicular and the axis. The final figure shows the mirror symmetry of the shapes about the line.
The prompt above was compiled from the original drawing commands and is for reference only.