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Distance from Center to Vertices of Dilated Triangles (Similar Triangles)

Distance from Center to Vertices of Dilated Triangles (Similar Triangles)
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2016-11-29 07:30
TriangleTransformationGeometric ConstructionSlider Animation
①This example dynamically demonstrates how the distances from the center to the vertices of a triangle change after a dilation transformation, visually revealing the geometric relationship between corresponding points and the center in similar triangles. ②Core mathematical concepts include dilation, similar triangles, proportional relationships, and distance measurement, involving rotation and the Pythagorean theorem. ③Construction highlights: using GeoGebra commands such as Polygon to create the original triangle, Dilate to scale it about a center with a scale factor, and Angle, Line, Segment to annotate angles and connecting lines. Multiple objects (A, B, C, poly1, D, SF, E) and their derived versions (A', B', C', etc.) illustrate different scale factors. ④Suitable for upper middle school or high school students to understand properties of dilation, correspondence in similar triangles, and the connection between distance and ratio in dynamic geometry; also useful for teacher demonstrations or student self-exploration.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Dilate(A, SF, E) → A'
  3. Dilate(B, SF, E) → B'
  4. Dilate(C, SF, E) → C'
  5. Polygon(A', B', C') → poly1', c', a', b'
  6. Angle(poly1) → α, β, γ
  7. Angle(poly1') → δ, ε, ζ
  8. Line(E, A) → f
  9. Line(E, B) → g
  10. Line(E, C) → h
  11. Polygon(A_1, B_1, C_1) → poly1_1, c_1, a_1, b_1
  12. Angle(poly1_1) → α_1, β_1, γ_1
  13. Polygon(A_2, B_2, C_2) → poly1_2, c_2, a_2, b_2
  14. Angle(poly1_2) → α_2, β_2, γ_2
  15. Polygon(A_3, B_3, C_3) → poly1_3, c_3, a_3, b_3
  16. Angle(poly1_3) → α_3, β_3, γ_3
  17. Segment(E, B) → eb
  18. Segment(E, B') → eb'
  19. Segment(E, A) → ea
  20. Segment(E, A') → ea'
  21. Segment(E, C) → ec
  22. Segment(E, C') → ec'

AI Drawing Prompt Reference

Draw a triangle ABC, then perform a similarity transformation around a center point E by changing the scale factor SF to observe the changes in distances from the center to the vertices of the triangle before and after the transformation. First, draw triangle ABC, then use scale factor SF to create a similar triangle A'B'C' around point E. Next, label the interior angles of triangles ABC and A'B'C', and draw lines connecting point E to each vertex. Finally, measure and display the lengths of these lines.
The prompt above was compiled from the original drawing commands and is for reference only.