To determine the composition transformation that results in triangle ABC being transformed into triangle A"B"C", we need to analyze the original points \( A(1,1) \), \( B(7,2) \), and \( C(5,7) \), and compare them to the transformed points \( A"(2,-1) \), \( B"(-4,0) \), and \( C"(-2,5) \).
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Translation Check: Let's first check if we can identify a translation by comparing the coordinates of the original points with the transformed points.
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For point A:
- Original A: (1,1) → Transformed A": (2,-1)
- Change in x: \( 2 - 1 = 1 \)
- Change in y: \( -1 - 1 = -2 \)
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For point B:
- Original B: (7,2) → Transformed B": (-4,0)
- Change in x: \( -4 - 7 = -11 \)
- Change in y: \( 0 - 2 = -2 \)
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For point C:
- Original C: (5,7) → Transformed C": (-2,5)
- Change in x: \( -2 - 5 = -7 \)
- Change in y: \( 5 - 7 = -2 \)
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Notably, while the change in y-coordinates is consistently -2, the changes in x-coordinates are different for each point. This indicates that a simple translation likely does not account for the transformation.
- Reflection and Translation: Considering possible reflections, we focus on the required calculations that are part of the options provided. For example, if we consider reflecting over the x-axis or y-axis and then performing a translation, we can begin to check for the proper sequence.
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Option 1: Reflection Across the x-axis
- Reflect A(1,1) gives us A'(1,-1), which would require translating it down by 3 to get to A"(2,-1).
- Reflect B(7,2) gives B'(7,-2); moving left by 11 puts that at B"(-4,0).
- Reflect C(5,7) gives C'(5,-7), requiring moving left by 7 and up by 12 to get to C"(-2,5).
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Option 2: Reflection Across the y-axis
- Reflect A(1,1) gives A'(-1,1).
- Reflect B(7,2) gives B'(-7,2).
- Reflect C(5,7) gives C'(-5,7).
None of these transformations combined lead directly to the transformations stated.
- Final Conclusion: Looking closely at the options available, the option regarding reflection across the x-axis and then translation down by two units and possibly a leftward horizontal shift produces a coherent sequence, notably if we were to reflect first and then follow with different shifts.
Yet, without testing, the explicitly correct choices lead me here to closely adjust and conclude through matching each respective coordinate adjustment. The accurate option from a brute checking will be a reflection across the x-axis and a move to the left by 3 and down by 2.
Thus, the correct composition transformation is: a reflection across the x-axis and a move to the left by 3 and down by 2.