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Squares on a Right-Angled Triangle

Squares on a Right-Angled Triangle
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2017-03-27 03:19
TriangleTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates constructing squares on the sides of a right-angled triangle and exploring the relationships between these squares and the original triangle. Core mathematical concepts include the Pythagorean Theorem, trigonometric functions (sine, cosine), vectors, and coordinate geometry. Construction highlights involve GeoGebra commands like Point, Intersect, Polygon, Midpoint, Length, and Sequence. Suitable for high school mathematics education to understand geometric proofs of the Pythagorean Theorem and applications of trigonometric functions.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. Point(xAxis) → C
  4. Polygon(A, B, C) → poly1, c, a, b
  5. Polygon(A, C, 4) → poly2, f, g, h, i, D, E
  6. Polygon(B, A, 4) → poly3, j, k, l, m, F, G
  7. Polygon(C, B, 4) → poly4, n, p, q, r, H, I
  8. Midpoint(F, B) → J
  9. Midpoint(B, I) → K
  10. Midpoint(C, E) → L
  11. Midpoint(A, B) → M
  12. Midpoint(B, C) → N
  13. Midpoint(C, A) → O
  14. Length(b) → LHyp
  15. Sequence(A + (t * (C - A)), t, 0, 1, 1 / LHyp) → HypB
  16. Sequence(E + (t * (D - E)), t, 0, 1, 1 / LHyp) → HypT
  17. Sequence(A + (t * (E - A)), t, 0, 1, 1 / LHyp) → HypL
  18. Sequence(C + (t * (D - C)), t, 0, 1, 1 / LHyp) → HypR
  19. Sequence(Segment[Element[HypL, k], Element[HypR, k]], k, 1, LHyp) → HypLR
  20. Sequence(Segment[Element[HypT, k], Element[HypB, k]], k, 1, LHyp) → HypTB
  21. Length(c) → LS1
  22. Length(a) → LS2
  23. Sequence(B + (t * (A - B)), t, 0, 1, 1 / LS1) → S1B
  24. Sequence(G + (t * (F - G)), t, 0, 1, 1 / LS1) → S1T
  25. Sequence(A + (t * (F - A)), t, 0, 1, 1 / LS1) → S1R
  26. Sequence(B + (t * (G - B)), t, 0, 1, 1 / LS1) → S1L
  27. Sequence(B + (t * (C - B)), t, 0, 1, 1 / LS2) → S2B
  28. Sequence(H + (t * (I - H)), t, 0, 1, 1 / LS2) → S2T
  29. Sequence(C + (t * (I - C)), t, 0, 1, 1 / LS2) → S2L
  30. Sequence(B + (t * (H - B)), t, 0, 1, 1 / LS2) → S2R
  31. Sequence(Segment[Element[S1L, k], Element[S1R, k]], k, 1, LS1) → S1LR
  32. Sequence(Segment[Element[S2L, k], Element[S2R, k]], k, 1, LHyp) → S2LR
  33. Sequence(Segment[Element[S1T, k], Element[S1B, k]], k, 1, LS1) → S1TB
  34. Sequence(Segment[Element[S2T, k], Element[S2B, k]], k, 1, LS2) → S2TB