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Similar Right Triangles Animation: Altitude on Hypotenuse

Similar Right Triangles Animation: Altitude on Hypotenuse
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2020-04-23 13:37
TriangleTransformationGeometric ConstructionSlider Animation
In collections Geometry - Mini Similarity Book
① This example demonstrates, through an animated visualization, how the altitude drawn to the hypotenuse of a right triangle divides the original triangle into two smaller right triangles that are each similar to the original. ② The core mathematical concepts are triangle similarity (AA criterion) and the geometric mean (projection) relations in a right triangle with an altitude to the hypotenuse. ③ Construction highlights: using GeoGebra commands such as Line, Intersect, OrthogonalLine, Segment, Polygon, Ray, Vector, Translate, Angle, AngularBisector, Rotate, Midpoint, and Mirror to build the right triangle, the altitude, the subdivided triangles, and their translated/rotated copies; a slider or animation drives the dynamic transformation. ④ Suitable for middle-to-high school geometry classes to explore similarity properties, the geometric mean theorem, and dynamic transformation in geometry.

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 with an altitude drawn from the right angle to the hypotenuse, dividing it into two smaller similar right triangles. Animate the altitude and show the two smaller triangles rotating to fit onto the original triangle, highlighting the concept of triangle similarity and geometric mean.
The prompt above was compiled from the original drawing commands and is for reference only.