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Bank Shot via Mirror Reflection Construction: Solve by Equal Angles

Bank Shot via Mirror Reflection Construction: Solve by Equal Angles
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2021-01-19 16:36
CircleAngleTransformationGeometric ConstructionGraphing
This example demonstrates the “bank shot” principle: when a path reflects off a line (the mirror/reflecting surface) from a start point to a target point, the reflection condition can be solved using the mirror method. Instead of tracking the reflected trajectory directly, you construct an appropriate mirror image of the target and connect it with a straight segment, yielding the correct rebound point and direction. Key mathematics includes the equal-angles (law of reflection), geometric mirror transformation, construction of segments/rays, and the use of intersections to locate the reflection point. Construction workflow is built from GeoGebra commands such as Segment and Point for the main geometric entities, Angle to visualize and compare the relevant angles at the bounce, Mirror to generate image points, Ray to represent outgoing directions, and Intersect to compute the bounce location where the path meets the reflecting line. Display/annotation objects (e.g., toggling segments/paths and showing angles and labels) help verify the reflection constraints. Suitable for middle school through early university geometry instruction, especially for teaching reflection laws and geometric construction techniques.

GeoGebra Commands & Parameters

  1. Segment(D, E) → f
  2. Point(f) → C
  3. Point(h) → A
  4. Point(g) → B
  5. Segment(A, C) → i
  6. Segment(C, B) → j
  7. Angle(E, C, B) → α
  8. Angle(A, C, D) → β
  9. Mirror(A, k) → A'
  10. Segment(A, C) → l
  11. Ray(C, A') → m
  12. Intersect(g, m) → F
  13. Angle(E, C, A') → γ
  14. Angle(A, C, D) → δ
  15. Segment((-10, -10), (-11, -11)) → p
  16. Segment(C, F) → n

AI Drawing Prompt Reference

Draw a horizontal reflecting line. Place a bounce point on it. Put a start point and a target point on the same side, each lying on a separate auxiliary line. Connect the start to the bounce and the bounce to the target with segments, and mark the two angles these segments make with the reflecting line. Reflect the target across the line to get its mirror image. From the bounce point, draw a ray toward the mirror image; its intersection with the target's auxiliary line gives the actual rebound point. Connect the bounce to this rebound point)Skip, and show equal angle marks at the bounce to verify the law of reflection.
The prompt above was compiled from the original drawing commands and is for reference only.