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Right Triangle Similarity

Right Triangle Similarity
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2017-04-01 00:31
TriangleAngleVectorTransformationGeometric Construction
In collections Unit 9: Right Triangles and Trigonometry
This example demonstrates the concept of similarity in right triangles. By constructing right triangles and their similar counterparts, it illustrates the properties of similar triangles. It uses commands such as Line, Intersect, OrthogonalLine, Segment, Polygon, Ray, Vector, Translate, Angle, AngularBisector, Rotate, Midpoint, Mirror in GeoGebra. Suitable for middle school and above students to learn about similar triangles.

GeoGebra Commands & Parameters

  1. Line(PP, C) → a_1
  2. Intersect(a_1, xAxis) → A
  3. OrthogonalLine(C, a_1) → b_1
  4. Intersect(b_1, xAxis) → B
  5. Segment(C, A) → b
  6. Segment(C, B) → a
  7. Segment(A, B) → c
  8. OrthogonalLine(C, c) → d
  9. Intersect(d, c) → D_1
  10. Segment(C, D_1) → e
  11. Polygon(C, D_1, B) → poly1, b_2, c_1, d_1
  12. Polygon(C, D_1, A) → poly2, a_2, c_2, d_2
  13. Ray(A, C) → f
  14. Ray(A, B) → g
  15. Ray(C, B) → h
  16. Vector(A, C') → u
  17. Translate(C, u) → C'_1
  18. Translate(D_1, u) → D'
  19. Translate(A, u) → A'
  20. Polygon(C'_1, D', A') → poly2', a', c'_1, d'
  21. Vector(C, B') → v
  22. Translate(C, v) → C'_2
  23. Translate(D_1, v) → D'_1
  24. Translate(B, v) → B'_1
  25. Polygon(C'_2, D'_1, B'_1) → poly1', b'_1, c'_2, d'_1
  26. Polygon(A, C, B) → poly3, b_3, a_3, c_3
  27. Vector(A, AA) → w
  28. Translate(A, w) → A'_1
  29. Translate(C, w) → C'_3
  30. Translate(B, w) → B'_2
  31. Polygon(A'_1, C'_3, B'_2) → poly3', b'_2, a'_1, c'_3
  32. Angle(A'_1, C'_3, B'_2) → α
  33. Angle(B'_2, A'_1, C'_3) → β
  34. Angle(C'_3, B'_2, A'_1) → γ
  35. Angle(C, B, A) → δ
  36. AngularBisector(A', C'_1, D') → j
  37. AngularBisector(C'_1, D', A') → k
  38. Intersect(j, k) → E
  39. Rotate(C'_1, (n * (180° - β)), E) → F
  40. Rotate(D', (n * (180° - β)), E) → D''
  41. Rotate(A', (n * (180° - β)), E) → A''
  42. Polygon(F, D'', A'') → poly2'', a'', f_1, d''
  43. AngularBisector(D'_1, C'_2, B'_1) → l
  44. AngularBisector(C'_2, B'_1, D'_1) → m
  45. Intersect(l, m) → G
  46. Angle(A'_1, B, C) → ε
  47. Rotate(C'_2, (-((n * ε))), G) → H
  48. Rotate(D'_1, (-((n * ε))), G) → D
  49. Rotate(B'_1, (-((n * ε))), G) → J
  50. Polygon(H, D, J) → poly1'', j_1, h_1, i_1
  51. Midpoint(d'') → I
  52. Midpoint(i_1) → K
  53. OrthogonalLine(I, d'') → p
  54. OrthogonalLine(K, i_1) → q
  55. Mirror(F, p) → F'
  56. Mirror(D'', p) → D'''
  57. Mirror(A'', p) → A'''
  58. Polygon(F', D''', A''') → poly2''', a''', f', d'''
  59. Mirror(H, q) → H'
  60. Mirror(D, q) → D'_2
  61. Mirror(J, q) → J'
  62. Polygon(H', D'_2, J') → poly1''', j', h', d'_2
  63. Angle(A, C, B) → ζ
  64. Angle(C'_1, D', A') → η
  65. Angle(B'_1, D'_1, C'_2) → θ
  66. Angle(D_1, A, C) → ι
  67. Angle(D', A', C'_1) → κ
  68. Segment(A, D_1) → r
  69. Segment(D_1, B) → s
  70. Angle(D'', A'', F) → λ
  71. Angle(F, D'', A'') → μ
  72. Angle(C, D_1, A) → ν
  73. Angle(B, D_1, C) → ξ
  74. Angle(A''', D''', F') → ο
  75. Angle(F', A''', D''') → ρ
  76. Angle(C'_2, B'_1, D'_1) → σ
  77. Angle(J, D, H) → τ
  78. Angle(H, J, D) → υ
  79. Angle(H', D'_2, J') → ϕ
  80. Angle(D'_2, J', H') → χ

AI Drawing Prompt Reference

Draw a right triangle ABC with C as the right angle. Construct a perpendicular line from C to AB, intersecting at D. Connect CD, AB, and BC to form two right triangles. Then, translate and rotate triangle ABC to create similar triangles A'B'C' and A''B''C''. Label angles α, β, γ, and δ. Finally, find the intersection points of the angle bisectors of triangles A'B'C' and A''B''C'', rotate to get new triangle F'D'A''.
The prompt above was compiled from the original drawing commands and is for reference only.