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Similar Right Triangles (II)

Similar Right Triangles (II)
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2016-03-02 10:40
TriangleCircleTrigonometryGeometric Construction
This example demonstrates the phenomenon where two smaller right triangles formed by drawing an altitude to the hypotenuse of a right triangle are similar to each other and to the original triangle. It involves core math concepts like similarity criteria for triangles, properties of right triangles, and trigonometric functions. The construction uses GeoGebra commands such as Segment, Midpoint, OrthogonalLine, Circle, Semicircle, Point, Intersect, Angle, Polygon, CircleSector, Dilate, Vector, Translate, Rotate, CircleArc. It is suitable for high school students to understand the similarity theorem and its practical applications.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Midpoint(a) → C_1
  3. OrthogonalLine(C_1, a) → b
  4. Circle(C_1, A) → c
  5. Semicircle(A, B) → d
  6. Point(d) → C
  7. Segment(A, C) → e
  8. Segment(C, B) → f
  9. OrthogonalLine(C, a) → g
  10. Intersect(g, a) → E
  11. Segment(C, E) → h
  12. Angle(B, E, C) → α
  13. Angle(A, C, B) → β
  14. Polygon(A, E, C) → poly1, d_1, a_1, e_1
  15. Polygon(E, B, C) → poly2, d_2, e_2, b_1
  16. Circle(A, 1) → k
  17. Circle(C, 1) → p
  18. Circle(B, 1) → q
  19. Intersect(p, e_1, 1) → F
  20. Intersect(a_1, b_1) → G
  21. Intersect(a_1, b_1) → H
  22. Intersect(p, h, 1) → I
  23. Intersect(p, f, 1) → J
  24. Intersect(q, a, 1) → K
  25. Intersect(q, f, 1) → L
  26. Intersect(k, a, 1) → M
  27. Intersect(k, e, 1) → N
  28. CircleSector(A, M, N) → r
  29. Dilate(C, If[-2 < i < -1, -1 - i, 0], E) → D'
  30. Segment(C, D') → j
  31. CircleSector(C, I, J) → s
  32. CircleSector(B, L, K) → t
  33. CircleSector(C, F, I) → c_1
  34. Angle(C, E, A) → γ
  35. Dilate(A, If[-0.5 < i < 0.5, 0.5 - i, 1], C) → A'
  36. Vector(A, A') → u
  37. Translate(r, u) → r'
  38. Rotate(r', If[-0.5 < i < 0.5, (((-(i + 0.5)) * 90))°], A') → r''
  39. CircleArc(A, M, N) → f_1
  40. Dilate(B, If[0.5 < i < 1.5, 1.5 - i], C) → B'
  41. Vector(B, B') → v
  42. Translate(t, v) → t'
  43. Rotate(t', If[0.5 < i < 1.5, (((i - 0.5) * 90))°], B') → t''
  44. CircleArc(B, L, K) → g_1
  45. Polygon(A, B, C) → poly3, d_3, a_2, b_2