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Properties of Tangents to Circles Interactive Investigation

Properties of Tangents to Circles Interactive Investigation
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2019-06-29 05:41
CircleAngleCoordinate GeometryGeometric Construction
This example demonstrates how to explore the properties of tangents to circles through interactive operations. Key math concepts involved include tangent lines, circles, angles, similar triangles, and the Pythagorean theorem. Construction highlights the use of GeoGebra commands and objects such as Circle, Segment, PointIn, Tangent, Intersect, Ray, Dilate, Angle, Rotate, CircleSector, and Line. Suitable for high school students to understand geometric properties of circles and conduct exploratory learning.

GeoGebra Commands & Parameters

  1. Circle(O, B_1) → c
  2. Segment(O, B_1) → a
  3. PointIn(b) → E
  4. Tangent(E, c) → d, e
  5. Intersect(c, e, 1) → B
  6. Intersect(c, d, 1) → A
  7. Ray(E, A) → f
  8. Ray(E, B) → g
  9. Segment(O, A) → h
  10. Segment(O, B) → i
  11. Dilate(O, If[0 < j ≤ 1, 1 - j, 0], A) → A'
  12. Dilate(O, If[0 < j ≤ 1, 1 - j, 0], B) → A'_1
  13. Segment(O, A') → k
  14. Segment(O, A'_1) → l
  15. Angle(E, B, O) → α
  16. Angle(O, A, E) → β
  17. Circle(A, (0.25 * a)) → p
  18. Circle(B, (0.25 * a)) → q
  19. Intersect(p, h, 1) → F
  20. Intersect(q, i, 1) → G
  21. Intersect(p, f) → H, I
  22. Intersect(q, g) → J, K
  23. Circle(O, B_1) → t
  24. Dilate(O, If[3.6 < j ≤ 4.6, 4.6 - j, 0], E) → C'
  25. Angle(A, E, B) → γ
  26. Rotate(B, If[5 < j ≤ 6, ((j - 5) * ((2 * π) - γ)), (2 * π) - γ], E) → D'_1
  27. Rotate(E, If[5 < j ≤ 6, ((j - 5) * ((2 * π) - γ)), (2 * π) - γ], E) → C'_1
  28. Segment(E, A) → n
  29. Ray(A, I) → m
  30. Ray(B, K) → g_1
  31. Rotate(E, If[5 < j ≤ 6, ((j - 5) * (-γ)), (-γ)], E) → C'_2
  32. Rotate(B, If[5 < j ≤ 6, ((j - 5) * (-γ)), (-γ)], E) → D'
  33. Rotate(I, If[1.5 < j ≤ 2.5, ((j - 1.5) * π / 2), π / 2], A) → I'
  34. Rotate(K, If[1.5 < j ≤ 2.5, ((-(j - 1.5)) * π / 2), ((-π)) / 2], B) → K'
  35. CircleSector(A, I, I') → r
  36. CircleSector(B, K', K) → s
  37. Rotate(r, If[3.25 < j ≤ 4.25, ((j - 3.25) * π / 2), π / 2], A) → r'
  38. Rotate(s, If[3.25 < j ≤ 4.25, ((-(j - 3.25)) * π / 2), ((-π)) / 2], B) → s'
  39. Dilate(E, If[-1 < j ≤ 0, (-j), 0], A) → E'
  40. Dilate(E, If[-1 < j ≤ 0, (-j), 0], B) → E'_1
  41. Segment(E, E'_1) → f_1
  42. Segment(E, E') → i_1
  43. Line(E, A) → j_1
  44. Line(E, B) → k_1
  45. Circle(B, 2) → p_1
  46. Circle(A, 2) → q_1
  47. Intersect(q_1, j_1) → C, D
  48. Intersect(p_1, k_1) → L, M
  49. Ray(A, D) → l_1
  50. Ray(B, M) → m_1
  51. Rotate(E, If[4.75 < j ≤ 5.75, ((j - 4.75) * γ), γ], E) → E'_2
  52. Rotate(E', If[4.75 < j ≤ 5.75, ((j - 4.75) * γ), γ], E) → E''
  53. Segment(E'_2, E'') → n_1