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Dilating a Segment: Parallel and Scaled by Factor k (HSG.SRT.A.1.B)

Dilating a Segment: Parallel and Scaled by Factor k (HSG.SRT.A.1.B)
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2016-02-26 00:53
VectorTransformationGeometric ConstructionSlider Animation
In collections Geometry Review Similarity & Right Triangle Trigonometry Similarity, Right Triangles, and Trigonometry (SRT) Cluster Transformations! (Geometry) Dilations: Discovery Activities & Exercises Dilations: Discovery Activities & Exercises
This applet demonstrates that when a line segment is dilated about a center point with scale factor k, its image is another segment that is parallel to the original and has length k times the original. The core mathematical concept is dilation (a similarity transformation) and its properties: parallelism of corresponding segments, proportional lengths, and the effect of the scale factor on position and size. Construction details: using GeoGebra commands such as Line, Dilate, Intersect, Circle, CircleSector, Angle, Vector, Translate, Segment, Point, and Ray, the applet builds center O, a slider for scale factor k, original segment BC and its image B'C', with auxiliary circles, angles, and vectors to verify parallelism and length ratio. This resource is suitable for high school geometry students learning similarity transformations and the Common Core standard HSG.SRT.A.1.b, and can be used for classroom demonstration or independent exploration.

GeoGebra Commands & Parameters

  1. Line(B, C) → m
  2. Dilate(m, k, O) → m'
  3. Intersect(m, t) → O_1
  4. Line(O, C) → f
  5. Circle(C, 2) → g
  6. Intersect(m', f) → E
  7. Circle(E, 2) → h
  8. Intersect(g, m, 2) → F
  9. Intersect(g, f, 2) → G
  10. Intersect(h, m', 2) → H
  11. Intersect(h, f, 2) → I
  12. Intersect(g, m, 1) → J
  13. Intersect(g, f, 1) → K
  14. CircleSector(C, F, G) → q
  15. Angle(F, C, G) → α
  16. Angle(J, C, G) → β
  17. CircleSector(C, J, G) → c
  18. Intersect(h, m', 1) → N
  19. Dilate(C, 1 - i, E) → C'_3
  20. Vector(C, C'_3) → u
  21. Translate(q, u) → q'
  22. Translate(c, u) → c'
  23. Circle(O, 2) → e
  24. Segment(B, C) → preimage
  25. Dilate(B, k, O) → B'_2
  26. Dilate(C, k, O) → C'_1
  27. Segment(B'_2, C'_1) → image
  28. Dilate(B, k, O) → B'
  29. Dilate(C, k, O) → C'
  30. Point(m') → A
  31. Ray(C', A) → j
  32. Point(j) → D
  33. Point(m) → P
  34. Ray(C, P) → l
  35. Point(l) → Q

AI Drawing Prompt Reference

Draw a dilation transformation centered at point O with a scale factor k controlled by a slider. Start with a line through points B and C, then dilate this line about O to obtain a parallel line. Mark the original segment BC and its image B'C' on these lines. Add auxiliary lines from O to C and O to C', and use circles and circular sectors to indicate corresponding angles, demonstrating that the image segment is parallel to the original and its length is k times the original. Include a vector arrow showing the translation from the original segment to the image, and add auxiliary points to clarify the geometric relationships.
The prompt above was compiled from the original drawing commands and is for reference only.