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Properties of Tangents Drawn to Circles (A) – Interactive Student Investigation

Properties of Tangents Drawn to Circles (A) – Interactive Student Investigation
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2016-04-22 06:49
CircleAngleTransformationGeometric ConstructionSlider Animation
In collections Circles (Theorems)
① This interactive GeoGebra example demonstrates the key properties of tangents drawn from an external point to a circle, such as equal tangent lengths and the perpendicularity between the tangent and the radius at the point of tangency. ② Core mathematical concepts include tangent lines, points of tangency, the tangent-length theorem, perpendicular relationships, angle bisectors, and properties of quadrilaterals formed by tangent points. ③ Construction highlights: using GeoGebra commands like Circle, Tangent, Intersect, Segment, Ray, Angle, Rotate, Dilate, and a button (button1) for interactivity, the applet lets users drag points to explore dynamic changes in the figure. ④ Suitable for middle or high school geometry classes, this resource serves both as a teacher demonstration tool and a student-led investigation for understanding tangent properties and their geometric proofs.

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