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Measuring Polygon Area with Vector Translation

Measuring Polygon Area with Vector Translation
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2019-04-02 01:09
PolygonVectorMatrixTransformationGeometric Construction
① This example demonstrates how to measure and display the area of a polygon dynamically using vector translation and piecewise functions. ② Core mathematical concepts include polygon area calculation, vector translation, piecewise functions, and the use of matrices and dilation in geometric construction. ③ Construction highlights: GeoGebra commands such as Segment, Vector, Translate, and Polygon are used to build the polygon and its translated copies; objects like t1, t2, poly1, A', B', E' are linked, with text objects showing the area value, while matrices and cubes serve as auxiliary elements. ④ Suitable for middle or high school geometry classes or math lab activities, helping students intuitively understand area invariance under translation and the geometric meaning of vectors.

GeoGebra Commands & Parameters

  1. Segment(X0, X0 + (0, 0.25)) → g
  2. Segment(X0, X0 + (0, mark)) → h
  3. Segment(X0 + (1, 0), X0 + (1, mark)) → i
  4. Vector(X0 + (0, mark / 2), X0 + (1, mark / 2)) → u
  5. Vector(X0 + (1, mark / 2), X0 + (0, mark / 2)) → v
  6. Segment(Y0 - (mark, 0), Y0) → j
  7. Segment(Y0 - (mark, 1), Y0 - (0, 1)) → k
  8. Vector(Y0 - (mark / 2, 1), Y0 - (mark / 2, 0)) → w
  9. Vector(Y0 - (mark / 2, 0), Y0 - (mark / 2, 1)) → z
  10. Segment((3, -1), (3, 3)) → l
  11. Translate(A, Vector[(t1 * Vector[A, D])]) → A'
  12. Translate(B, Vector[(t1 * Vector[A, D])]) → B'
  13. Translate(E, Vector[(t1 * Vector[A, D])]) → E'
  14. Polygon(A', B', E') → poly1, e', a', b'
  15. Translate(B, Vector[(t2 * Vector[B, A])]) → B'_1
  16. Translate(C, Vector[(t2 * Vector[B, A])]) → C'
  17. Translate(E, Vector[(t2 * Vector[B, A])]) → E'_1
  18. Polygon(B'_1, C', E'_1) → poly2, e'_1, b'_1, c'
  19. Translate(C, Vector[(t3 * Vector[D, A])]) → C'_1
  20. Translate(D, Vector[(t3 * Vector[D, A])]) → D'
  21. Translate(E, Vector[(t3 * Vector[D, A])]) → E'_2
  22. Polygon(C'_1, D', E'_2) → poly3, e'_2, c'_1, d'
  23. Translate(D, Vector[(t4 * Vector[A, B])]) → D'_1
  24. Translate(E, Vector[(t4 * Vector[A, B])]) → E'_3
  25. Translate(A, Vector[(t4 * Vector[A, B])]) → A'_1
  26. Polygon(D'_1, E'_3, A'_1) → poly4, a'_1, d'_1, e'_3
  27. Polygon(A', B', E') → poly1', e'_4, a'_2, b'_2
  28. Polygon(B'_1, C', E'_1) → poly2', e'_5, b'_3, c'_2
  29. Polygon(C'_1, D', E'_2) → poly3', e'_6, c'_3, d'_2
  30. Polygon(D'_1, E'_3, A'_1) → poly4', a'_3, d'_3, e'_7
  31. Polygon(A, B, C, D) → poly5, a, b, c, d
  32. Polygon(A, B, E) → poly6, e, a_1, b_1
  33. Polygon(B, C, E) → poly7, e_1, b_2, c_1
  34. Polygon(C, D, E) → poly8, e_2, c_2, d_1
  35. Polygon(A, D, E) → poly9, e_3, a_2, d_2

AI Drawing Prompt Reference

Draw a quadrilateral with an interior point connected to its four vertices, dividing it into four triangles. Translate each triangle along the direction of one side of the quadrilateral, aligning a vertex with the midpoint of an adjacent side, creating four translated copies. Add arrows to indicate the translation vectors and display the area of each triangle to show that area is preserved under translation.
The prompt above was compiled from the original drawing commands and is for reference only.