To find the composition transformation rule that maps triangle \( \triangle LMN \) with vertices \( L(1,1) \), \( M(7,2) \), and \( N(5,7) \) onto triangle \( \triangle L"M"N" \) with vertices \( L"(2,-1) \), \( M"(-4,0) \), and \( N"(-2,5) \), we can analyze the transformations step by step.
Let's first look at the coordinates:
- \( L(1, 1) \) to \( L"(2, -1) \)
- \( M(7, 2) \) to \( M"(-4, 0) \)
- \( N(5, 7) \) to \( N"(-2, 5) \)
Step 1: Determine if there is a reflection
Let's determine if the transformation involves a reflection by checking how the points change.
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For point \( L \): from \( (1,1) \) to \( (2,-1) \):
- The change in the x-coordinate is \( 2 - 1 = 1 \) (moves right by 1).
- The change in the y-coordinate is \( -1 - 1 = -2 \) (moves down by 2).
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For point \( M \): from \( (7,2) \) to \( (-4,0) \):
- The change in the x-coordinate is \( -4 - 7 = -11 \) (moves left by 11).
- The change in the y-coordinate is \( 0 - 2 = -2 \) (moves down by 2).
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For point \( N \): from \( (5,7) \) to \( (-2,5) \):
- The change in the x-coordinate is \( -2 - 5 = -7 \) (moves left by 7).
- The change in the y-coordinate is \( 5 - 7 = -2 \) (moves down by 2).
Step 2: Determine the nature of the reflection
Considering that the points exhibit a significant change in the x-coordinates for \( M \), particularly, we can see that reflecting across the y-axis (which would negate the x-coordinates) could also explain the large movements.
Reflecting across the y-axis:
- \( L(1, 1) \) becomes \( L'(-1, 1) \)
- \( M(7, 2) \) becomes \( M'(-7, 2) \)
- \( N(5, 7) \) becomes \( N'(-5, 7) \)
Then adding a translation of \( (+3, -2) \):
- \( L'(-1, 1) \) becomes \( L"(2, -1) \)
- \( M'(-7, 2) \) becomes \( M"(-4, 0) \)
- \( N'(-5, 7) \) becomes \( N"(-2, 5) \)
This matches perfectly with the coordinates of the transformed triangle.
Conclusion
The transformations confirm that:
- Reflection across the y-axis
- Translation 3 units to the right and 2 units down
Thus, the correct answer to the composition transformation rule is:
A. a reflection across the y-axis and a move to the right by 3 and down by 2.