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Right Triangle Altitude Similarity

Right Triangle Altitude Similarity
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2015-01-23 05:50
TriangleVectorGeometric Construction
In collections Right Triangles and Trig
This example demonstrates the similarity of altitudes in a right triangle. Key mathematical concepts include the properties of similar triangles. Using GeoGebra commands such as Line, Intersect, OrthogonalLine, etc., the construction of a right triangle and its altitudes is illustrated. 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 the altitudes BD and CE from C to AB and from A to BC, respectively. Then, create similar triangles by translating and rotating the sides of the original triangle, ensuring the new triangles are similar to the original. Finally, label the angles and angle bisectors of these triangles.
The prompt above was compiled from the original drawing commands and is for reference only.