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Ambiguous Case of the Sine Law (SSA) Dynamic Exploration

Ambiguous Case of the Sine Law (SSA) Dynamic Exploration
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2018-12-18 01:53
TriangleAngleTrigonometryGeometric ConstructionSlider Animation
① This interactive GeoGebra example dynamically demonstrates the ambiguous case of the sine law, where given two sides and a non-included angle (SSA) can yield zero, one, or two possible triangles. ② The core mathematical concepts include the sine rule, triangle solvability conditions, and geometric intersection analysis. ③ Construction highlights: using GeoGebra commands such as Random, Min, Rotate, Point, Circle, Intersect, DynamicCoordinates, TableText, LaTeX, Polygon, Segment, and Angle to generate random parameters, display solution counts and angles in a table, and construct the triangle visually; sliders and buttons (e.g., TryAnother) enable interactive exploration. ④ Suitable for high school or pre-university students learning trigonometry and triangle solving, as well as for classroom demonstrations or self-study to grasp the multiplicity of solutions and their geometric origins.

GeoGebra Commands & Parameters

  1. Random(5, 13) → lenAB
  2. Min(lenAB - 1, (lenAB * sin(angleA)) + 3) → maxBC
  3. Random(minBC, maxBC) → lenBC
  4. Rotate((1, 0), angleA, A) → B'
  5. Point(Segment[(10 - lenBC, 0), (10 + lenBC, 0)]) → C_1
  6. Circle(B, lenBC) → Bcircle
  7. Point(Bcircle) → K_1
  8. Intersect(Bcircle, xAxis) → C'_1, C'_2
  9. DynamicCoordinates(K_1, If[Distance[C'_1, K_1] < 0.5, x(C'_1), If[Distance[C'_2, K_1] < 0.5, x(C'_2), x(K_1)]], If[Distance[C'_1, K_1] < 0.5, y(C'_1), If[Distance[C'_2, K_1] < 0.5, y(C'_2), y(K_1)]]) → C
  10. TableText({A1:A3, B1:B3, C1:C3, D1:D3, E1:E3}, "|10000 _100 v") → displayTbl
  11. LaTeX("C") → cLbl
  12. Point(Bcircle) → C''
  13. Polygon(A, B, C'_1) → t1, i, j, k
  14. Polygon(A, B, C'_2) → t2, f, g, h
  15. Segment(B, C) → swingArm
  16. Intersect(swingArm, xAxis) → K
  17. Angle(C'_1, A, B') → dispAngle
  18. Segment(A, A + (25, 0)) → bottomSegment
  19. LaTeX((((((("\text{\textbf{Question: }How many triangles can be constructed if \(\angle A =" * angleA) * "\), \(AB =") * lenAB) * "\), and \(BC =") * lenBC) * "\)?}")) → MainInstruc
  20. LaTeX("\text{\textbf{Instructions: }Drag point \(C\) to try to create as many different triangles as possible.}") → SpecInstruc

AI Drawing Prompt Reference

Draw a triangle with angle A randomly between 5 and 13 degrees, and side AB randomly between 5 and 13 units. Generate side BC randomly within the range of its minimum and maximum possible values. Rotate point B around point A by angle A to get point B'. On the x-axis, randomly place point C1, then draw a circle with center B and radius BC, and find the intersection points C'1 and C'2 with the x-axis. Choose the closer intersection point as point C to construct triangle ABC. Depending on the values of angle A, AB, and BC, you can construct 0, 1, or 2 different triangles.
The prompt above was compiled from the original drawing commands and is for reference only.