To determine the transformation that causes triangle \( \Delta ABC \) to become triangle \( \Delta DBC \), let's analyze the changes between these triangles.
The triangles share the common side \( BC \), and to transform triangle \( ABC \) into triangle \( DBC \), we need to focus on how point \( A \) moves to point \( D \). Since the triangles maintain side \( BC \), the transformation must involve repositioning point \( A \) to point \( D \).
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Dilation: A dilation would change the size of the triangle, which doesn't seem to apply here as there is no mention of size change; it's a rearrangement of points.
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Reflection: A reflection would create a mirror image across a line, which might change the orientation of the triangle. This could be a possibility if point \( D \) is the reflection of point \( A \) across the line \( BC \).
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Rotation: A rotation would involve turning the triangle around a point. If point \( D \) is positioned by rotating point \( A \) around point \( B \) or \( C \), it would also change the orientation, not simply move \( A \) to \( D \).
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Translation: A translation involves sliding the entire triangle in the plane. This could change the position of point \( A \) but would not create a new point \( D\) unless \( D \) represents a fixed point in the direction of the slide.
Based on this exploration, if \( D \) represents a position directly related to point \( A \) but shifted across the line \( BC \), the most likely transformation is:
Option #2: reflection
This transformation would cause triangle \( \Delta ABC \) to become triangle \( \Delta DBC \) by reflecting point \( A \) over the line that contains segment \( BC \).