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Unit Circle Tangent Function and Geometric Mean Exercise

Unit Circle Tangent Function and Geometric Mean Exercise
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2015-07-12 20:15
CirclePolygonFunctionTrigonometryGeometric Construction
In collections quadratic function
①This GeoGebra example constructs a comprehensive geometric figure combining the unit circle, tangent function, geometric mean, and perimeter, demonstrating the interconnections among these mathematical concepts through interactive construction. ②Core mathematical concepts include trigonometric functions on the unit circle (especially tangent), geometric mean (proportional mean), perimeter of polygons, and geometric relationships among points, lines, and segments. ③Construction highlights: from the summary, commands such as Line, OrthogonalLine, Point, Segment, Midpoint, Intersect, and Polygon are used, along with multiple text objects (text1 to text13) for labeling, and objects like poly1 (polygon), c_1, a_1, e_1, forming a dynamic adjustable model. ④Suitable for high school or pre-university students, this serves as a tool for deepening understanding of the link between trigonometric functions and geometric quantities, and can be used for classroom demonstrations or student self-exploration, offering visual insight into the geometric meaning of tangent on the unit circle and the manifestation of geometric mean in figures.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. OrthogonalLine(B, a) → b
  3. Point(b) → C
  4. Segment(A, C) → c
  5. Midpoint(A, C) → D
  6. OrthogonalLine(D, c) → d
  7. OrthogonalLine(C, b) → e
  8. Intersect(d, e) → E
  9. Polygon(A, E, C) → poly1, c_1, a_1, e_1

AI Drawing Prompt Reference

Draw a unit circle with its center at the origin and a fixed point on the right end of the horizontal diameter. From this point, draw a tangent line to the circlehol. Choose an arbitrary point on this tangent line, and connect it to the center with a segment. Find the midpoint of this segment, then draw a perpendicular line through this midpoint to the segment. Also draw a perpendicular line through the tangent point to the tangent line. These two perpendicular lines intersect at a point. Construct a right triangle using the center, the tangent point, and this intersection point as vertices. Label the sides and angles, and show the relationships among the tangent function, geometric mean, and the triangle's perimeter.
The prompt above was compiled from the original drawing commands and is for reference only.