MathPlotAI Logo MathPlotAI

Spherical Triangle: Special Right Triangle

Spherical Triangle: Special Right Triangle
Loading the math board and drawing, please wait…
2014-07-22 07:01
TrigonometrySolid GeometryGeometric Construction
In collections Spherical Geometry Ideas
This example demonstrates a special right spherical triangle with one 90-degree angle and another 60-degree angle, with the third angle ranging from 30 to 120 degrees. Using GeoGebra commands such as Sphere, PointIn, Plane, IntersectPath, Point, OrthogonalLine, Intersect, CurveCartesian, Line, Rotate, OrthogonalPlane, Angle, the properties of the spherical triangle can be dynamically explored. Suitable for high school and above students to study spherical geometry and trigonometry.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), 1) → a
  2. PointIn(a) → A
  3. PointIn(a) → B_1
  4. Plane(O, A, B_1) → b
  5. IntersectPath(b, a) → c
  6. Point(c) → B
  7. OrthogonalLine(O, b) → d
  8. Plane(A, d) → e
  9. IntersectPath(e, a) → f
  10. OrthogonalLine(O, e) → g
  11. Intersect(g, a, 2) → D
  12. Intersect(d, a, 2) → E
  13. CurveCartesian((x(A) * cos(t)) + (x(D) * sin(t)), (y(A) * cos(t)) + (y(D) * sin(t)), (z(A) * cos(t)) + (z(D) * sin(t)), t, 0, Angle[A, O, B]) → h
  14. Line(O, B) → i
  15. Rotate(A, -60*°, i) → A'
  16. Plane(A', i) → j
  17. IntersectPath(j, a) → k
  18. Intersect
  19. Intersect(f, k, 2) → C
  20. OrthogonalLine(O, j) → l
  21. Plane(C, l) → m
  22. OrthogonalLine(O, m) → n
  23. CurveCartesian((x(A) * cos(t)) + (x(E) * sin(t)), (y(A) * cos(t)) + (y(E) * sin(t)), (z(A) * cos(t)) + (z(E) * sin(t)), t, 0, Angle[C, O, A]) → p
  24. Line(O, A) → q
  25. OrthogonalPlane(A, q) → r
  26. IntersectPath(b, r) → s
  27. IntersectPath(e, r) → t
  28. Point(t) → G
  29. Point(s) → H
  30. Angle(H, A, G) → α
  31. OrthogonalLine(O, m) → a_1
  32. Intersect(a_1, a, 1) → F
  33. CurveCartesian((x(C) * cos(t)) + (x(F) * sin(t)), (y(C) * cos(t)) + (y(F) * sin(t)), (z(C) * cos(t)) + (z(F) * sin(t)), t, 0, Angle[C, O, B]) → o
  34. OrthogonalPlane(B, i) → u
  35. IntersectPath(b, u) → b_1
  36. IntersectPath(j, u) → c_1
  37. Point(c_1) → I
  38. Point(b_1) → J
  39. Angle(I, B, J) → β

AI Drawing Prompt Reference

Draw a special right spherical triangle with one 90-degree angle and another 60-degree angle, with the third angle ranging from 30 to 120 degrees. Start by creating a unit sphere and selecting three points A, B, and C on the sphere such that angle AOB is 90 degrees, angle BOC is 60 degrees, and angle COA varies from 30 to 120 degrees. Use rotation and translation to construct the three sides of the triangle and label the angles.
The prompt above was compiled from the original drawing commands and is for reference only.