Let's analyze the transformations step-by-step to determine the correct relationships between the segments and triangles.
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Dilation: ΔEFI is dilated by a scale factor of 1/2 with center F. This means that the points E and I (the vertices opposite F) are moved closer to F, reducing the lengths of the segments EF and IF to half their original lengths:
- \( FE' = \frac{1}{2} FE \)
- \( FI' = \frac{1}{2} FI \)
- The new points, E' and I', create a smaller triangle ΔEF'I'.
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Reflection: The triangle resulting from the dilation is then reflected over line a to create triangle ΔHFG. This reflection keeps the distances the same but flips the triangle.
After these transformations, we can analyze the relationship between the segments:
- Since F is the center of dilation and later serves as a vertex for triangle HFG, and considering the transformations performed:
- \( FH = 2 \times FE' \) (because segment FH is twice the reduced segment from the dilation)
- \( FG = 2 \times FI' \)
- Segment HG would equal \( EI' \) after dilation, as it is unchanged due to the reflection.
Using this understanding, we can summarize the relationships (in the form of ratios) in the triangles after applying both transformations.
Based on the options provided:
- The second option stating "two segment FH = segment FI, two segment FG = segment FE, and two segment HG = segment EI; ΔEFI ~ ΔGFH" reflects this understanding of the relationships.
However, option 3 and option 4 should also be correct interpretations but describing segment sizes differently may appear confusing based on context.
The most accurate statement describing the similarity and relationships derived from the transformations would be:
"Segment FH = two segment FI, segment FG = two segment FE, and segment HG = two segment EI; ΔEFI ~ ΔGFH."
So we conclude that the relationship is indeed properly characterized, confirming similar triangles due to proportional sides matching the respective transformations applied.
2 answers