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Explore Law of Cosines: Dynamic SAS Triangle

Explore Law of Cosines: Dynamic SAS Triangle
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2015-01-29 03:45
TriangleAngleTrigonometryGeometric ConstructionSlider Animation
This interactive GeoGebra example demonstrates the Law of Cosines in the Side-Angle-Side (SAS) configuration. By dragging vertices or adjusting the included angle, users can observe how the third side length and all interior angles update in real time, verifying the formula c² = a² + b² - 2ab·cos(γ). The core mathematical concepts include the Law of Cosines, triangle side-angle relationships, trigonometric functions, and geometric measurement. The construction uses GeoGebra commands such as Circle, Line, Intersect, Rotate, Angle, and Segment to build the triangle, with objects like IncludedAngle, SecondAngle, ThirdAngle, BottomSide, LeftSide, and UnknownSide, plus dynamic text labels (text1, text2, etc.) displaying measurements. Suitable for high school geometry or trigonometry classes, and useful for college prep review, this dynamic tool helps students grasp the geometric meaning behind the formula through hands-on manipulation.

GeoGebra Commands & Parameters

  1. Circle(IncludedAngle, Left) → c
  2. Circle(IncludedAngle, Bottom) → d
  3. Line(IncludedAngle, A) → a
  4. Intersect(d, a, 2) → SecondAngle
  5. Rotate(SecondAngle, Angle1, IncludedAngle) → B'
  6. Angle(SecondAngle, IncludedAngle, B') → α
  7. Line(IncludedAngle, B') → b
  8. Intersect(c, b, 2) → ThirdAngle
  9. Segment(IncludedAngle, SecondAngle) → BottomSide
  10. Segment(IncludedAngle, ThirdAngle) → LeftSide
  11. Segment(ThirdAngle, SecondAngle) → UnknownSide
  12. Angle(ThirdAngle, SecondAngle, IncludedAngle) → β
  13. Angle(IncludedAngle, ThirdAngle, SecondAngle) → γ
  14. Segment(ThirdAngle, SecondAngle) → e

AI Drawing Prompt Reference

Draw a triangle with two adjustable sides and an included angle to demonstrate the Law of Cosines. Start by drawing two sides and the angle between them, then construct circles using the sides as radii to find the third vertex. Connect the points to form the triangle, labeling all sides and angles dynamically to show measurements and verify the formula.
The prompt above was compiled from the original drawing commands and is for reference only.