To find the correct transformation that maps triangle \( \triangle LMN \) with vertices L(8,3), M(4,6), and N(5,9) onto triangle \( \triangle L'M'N' \) with vertices \( L'(-10,-3) \), \( M'(-6,0) \), and \( N'(-7,3) \), we can analyze the transformation step by step.
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Identify the changes in coordinates:
- Vertex L: From (8, 3) to (-10, -3)
- Change in x: \(-10 - 8 = -18\)
- Change in y: \(-3 - 3 = -6\)
- Vertex M: From (4, 6) to (-6, 0)
- Change in x: \(-6 - 4 = -10\)
- Change in y: \(0 - 6 = -6\)
- Vertex N: From (5, 9) to (-7, 3)
- Change in x: \(-7 - 5 = -12\)
- Change in y: \(3 - 9 = -6\)
- Vertex L: From (8, 3) to (-10, -3)
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Analyzing the transformations:
- It seems that all the vertices have undergone a vertical transformation that has affected the y-coordinates down by 6.
- For vertex L, the reflection must have occurred first to get the x-coordinates negative.
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Possible reflections:
- Reflecting over the y-axis changes the coordinates from (x, y) to (-x, y).
- Considering the reflection:
- L (8, 3) reflected over the y-axis becomes (-8, 3).
- After reflection, to move to L' (-10, -3), we need to move left by 2 and then down by 6.
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Validation against the options:
- The transformation can be summed up as follows:
- Reflection over the y-axis: (x, y) → (-x, y)
- Then translate them: move left by 2 and down by 6.
- The transformation can be summed up as follows:
Thus, the correct transformation rule is:
- A reflection over the y-axis and a move to the left by 2 and down by 6.
This matches the response provided:
- a reflection over the y-axis and a move to the left by 2 and down by 6.