MathPlotAI Logo MathPlotAI

Reflection over Non-Concurrent Lines

Reflection over Non-Concurrent Lines
Loading the math board and drawing, please wait…
2016-07-13 04:28
PolygonAngleVectorTransformationGeometric Construction
This example demonstrates the geometric transformation of reflecting a polygon over a set of non-concurrent lines (lines that do not all intersect at a single point). The core mathematical concepts include reflection (axial symmetry), properties of non-concurrent lines, and the congruence between the original and reflected figures. Construction details: using GeoGebra commands such as Line, Point, Polygon, Mirror, Segment, Distance, Intersect, Angle, Midpoint, Vector, and Translate to construct the original polygon, the reflection axes, and the reflected image, along with measurements and verifications. Suitable for middle and high school geometry classes to explore how reflection works when axes are not concurrent and how the position of the image changes accordingly.

GeoGebra Commands & Parameters

  1. Line(A, B) → R1
  2. Point(Circle[C, 5.548885023137532]) → D
  3. Polygon(C, D, E, F, G, H, I, J) → poly1, c, d, e, f, g, h, i, j
  4. Mirror(C, R1) → C'
  5. Mirror(D, R1) → D'
  6. Mirror(E, R1) → E'
  7. Mirror(F, R1) → F'
  8. Mirror(G, R1) → G'
  9. Mirror(H, R1) → H'
  10. Mirror(I, R1) → I'
  11. Mirror(J, R1) → J'
  12. Polygon(C', D', E', F', G', H', I', J') → poly1', c', d', e', f', g', h', i', j'
  13. Segment(C, C') → l
  14. Segment(D, D') → m
  15. Distance(C, C') → distanceCC'
  16. Distance(E, E') → distanceEE'
  17. Distance(F, F') → distanceFF'
  18. Line(K, L) → k
  19. Mirror(C', k) → C''
  20. Mirror(D', k) → D''
  21. Mirror(E', k) → E''
  22. Mirror(F', k) → F''
  23. Mirror(G', k) → G''
  24. Mirror(H', k) → H''
  25. Mirror(I', k) → I''
  26. Mirror(J', k) → J''
  27. Polygon(C'', D'', E'', F'', G'', H'', I'', J'') → poly1'', c'', d'', e'', f'', g'', h'', i'', j''
  28. Intersect(R1, k) → M_1
  29. Angle(C, M_1, C'') → β
  30. Angle(J, M_1, J'') → γ
  31. Angle(I, M_1, I'') → δ
  32. Angle(H, M_1, H'') → ε
  33. Angle(G, M_1, G'') → ζ
  34. Angle(F, M_1, F'') → η
  35. Angle(E, M_1, E'') → θ
  36. Angle(D, M_1, D'') → ι
  37. Point(R1) → P
  38. Point(k) → Q
  39. Line(P, Q) → f_1
  40. Mirror(C'', f_1) → C'''
  41. Mirror(D'', f_1) → D'''
  42. Mirror(E'', f_1) → E'''
  43. Mirror(F'', f_1) → F'''
  44. Mirror(G'', f_1) → G'''
  45. Mirror(H'', f_1) → H'''
  46. Mirror(I'', f_1) → I'''
  47. Mirror(J'', f_1) → J'''
  48. Polygon(C''', D''', E''', F''', G''', H''', I''', J''') → poly1''', c''', d''', e''', f''', g''', h''', i''', j'''
  49. Segment(C, C''') → g_1
  50. Midpoint(g_1) → R
  51. Segment(E, E''') → j_1
  52. Midpoint(j_1) → T
  53. Segment(D, D''') → i_1
  54. Midpoint(i_1) → S
  55. Segment(F, F''') → k_1
  56. Midpoint(k_1) → U
  57. Segment(G, G''') → l_1
  58. Midpoint(l_1) → V
  59. Segment(H, H''') → m_1
  60. Midpoint(m_1) → W
  61. Segment(I, I''') → n_1
  62. Midpoint(n_1) → Z
  63. Segment(J, J''') → p_1
  64. Midpoint(p_1) → A_1
  65. Line(S, U) → M
  66. Vector(N, O) → n
  67. Translate(C, n) → C'_1
  68. Translate(D, n) → D'_1
  69. Translate(E, n) → E'_1
  70. Translate(F, n) → F'_1
  71. Translate(G, n) → G'_1
  72. Translate(H, n) → H'_1
  73. Translate(I, n) → I'_1
  74. Translate(J, n) → J'_1
  75. Polygon(C'_1, D'_1, E'_1, F'_1, G'_1, H'_1, I'_1, J'_1) → poly1'_1, c'_1, d'_1, e'_1, f'_1, g'_1, h'_1, i'_1, j'_1
  76. Mirror(C'_1, M) → B_1
  77. Mirror(D'_1, M) → C_1
  78. Mirror(E'_1, M) → D_1
  79. Mirror(F'_1, M) → E_1
  80. Mirror(G'_1, M) → F_1
  81. Mirror(H'_1, M) → G_1
  82. Mirror(I'_1, M) → H_1
  83. Mirror(J'_1, M) → I_1
  84. Polygon(B_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1) → poly2, b_2, c_2, d_1, e_1, f_2, g_2, h_1, i_2

AI Drawing Prompt Reference

Draw a polygon and perform an axis symmetry transformation over two non-concurrent lines (k and R1) to generate the first symmetric polygon poly1'. Then, perform another axis symmetry transformation over a third line f_1 on the first symmetric polygon to generate the final symmetric polygon poly1'''. Observe the position changes of the polygon after each transformation.
The prompt above was compiled from the original drawing commands and is for reference only.