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Similar Triangles SSS Similarity Construction

Similar Triangles SSS Similarity Construction
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2018-09-24 21:24
TriangleGeometric Construction
This example demonstrates how to determine triangle similarity based on side lengths. The key mathematical concepts include the definition of similar triangles, the SSS similarity criterion, and geometric construction techniques. Construction involves using GeoGebra commands such as Rotate, Polygon, Intersect, etc., to create triangles and analyze side length relationships. Suitable for middle or high school mathematics teaching to understand the principles of similar triangles and construction methods.

GeoGebra Commands & Parameters

  1. Rotate((Random[30, 50] / 10, 0), Random[20, 100]°, A) → B
  2. Polygon(A, B, C) → t1, cLen, aLen, bLen
  3. If(ABmaxLenCase3 > BCmaxLenCase3, sqrt(ABmaxLenCase3^(2) - BCmaxLenCase3^(2)), x(TRviewBox) - x(A)) → ACmaxLenCase3
  4. Polygon(BLviewBox, (x(BLviewBox), y(TRviewBox)), TRviewBox, (x(TRviewBox), y(BLviewBox))) → viewBox, vbLeft, vbTop, vbRight, vbBottom
  5. Intersect(Ray[A, A + (ACmaxLenCase3, BCmaxLenCase3)], viewBox) → MaxB3InRegion
  6. Polygon(A, MaxB3InRegion, (x(TRviewBox), y(MaxB3InRegion)), (x(TRviewBox), y(A))) → restrictedB3Range, f, g, h, j
  7. If(ABmaxLenCase3 > ACmaxLenCase3, Intersect[Circle[A, ABmaxLenCase3], Ray[(ACmaxLenCase3, 0), (ACmaxLenCase3, 5)]], TRviewBox) → MaxB3
  8. If(aLen < cLen, restrictedB3Range, Polygon[(x(BLviewBox), y(A)), (x(BLviewBox), y(TRviewBox)), TRviewBox, (x(TRviewBox), y(A))]) → B3range
  9. PointIn(B3range) → B3
  10. Circle(A, Segment[A, B3]) → cAlenAB3
  11. UnitVector(Vector[A, B]) → uAB
  12. Point(Segment[A, Intersect[Ray[A, B], viewBox]]) → B4
  13. Line(B4, Vector[B, C]) → b4Line
  14. Intersect(Ray[A, C], b4Line) → C4'
  15. UnitVector(Vector[A, C]) → uAC
  16. Point(Segment[A, Intersect[Ray[A, C], viewBox]]) → C4
  17. Polygon(A, B4, C4') → t4b, t4b_c, t4b_a, t4b_b
  18. Line(C4, Vector[C, B]) → c4Line
  19. Intersect(Ray[A, B], c4Line) → B4'
  20. Polygon(A, B4', C4) → t4c, t4c_c, t4c_a, t4c_b
  21. Ray((ACmaxLenCase3, 0), (ACmaxLenCase3, 2)) → i
  22. Circle(B4, BClenCase3b) → cBlenBC4
  23. Circle(B3, BClenCase3b) → cBlenBC3
  24. Segment((C3start, y(A)), (C3end, y(A))) → C3range
  25. Point(C3range) → C3
  26. Circle(C3, BClenCase3b) → cC3lenBC
  27. Length(Segment[A, B3]) → ABlenCase3
  28. Intersect(cAlenAB3, cC3lenBC) → B3', B3'_1, ,
  29. Polygon(A, B3', C3) → t3c, t3c_c, t3c_a, t3c_b
  30. PointIn(viewBox) → B1
  31. Point(Segment[A, (x(TRviewBox), y(A))]) → C2
  32. Point(Segment[A, (x(TRviewBox), y(A))]) → C1
  33. Polygon(A, B1, C1) → tCase1, tCase1_c, tCase1_a, tCase1_b
  34. PointIn(viewBox) → B2
  35. Intersect(Circle[A, AClenCase2a], Ray[A, C]) → C2',
  36. Polygon(A, B2, C2') → tCase2b, tCase2b_c, tCase2b_a, tCase2b_b
  37. Intersect(Circle[A, ABlenCase2b], Ray[A, B2]) → B2',
  38. Polygon(A, B2', C2) → tCase2c, tCase2c_c, tCase2c_a, tCase2c_b
  39. If(bActive, If[caseNum ≟ 4, B4, If[caseNum ≟ 3, B3, If[caseNum ≟ 2, B2, B1]]], If[caseNum ≟ 4, B4', If[caseNum ≟ 3, B3', If[caseNum ≟ 2, B2', B1]]]) → B'lblPoint
  40. Intersect(cBlenBC3, Ray[A, C]) → C3'_1, C3'_2
  41. If(Defined[C3'_2], C3'_2, C3'_1) → C3'
  42. Polygon(A, B3, C3') → tCase3b, tCase3b_c, tCase3b_a, tCase3b_b
  43. If(bActive, If[caseNum ≟ 4, C4', If[caseNum ≟ 3, C3', If[caseNum ≟ 2, C2', C1]]], If[caseNum ≟ 4, C4, If[caseNum ≟ 3, C3, If[caseNum ≟ 2, C2, C1]]]) → C'lblPt

AI Drawing Prompt Reference

Draw three similar triangles by first selecting three points A, B, and C to form triangle ABC. Then, construct circles with A as the center and AB, AC as radii to find intersection points B3 and C3 with side BC. Use rotation and intersection methods to find points B4 and C4, and finally form three similar triangles: △AB3C3, △AB4C4, and △ABC.
The prompt above was compiled from the original drawing commands and is for reference only.