To determine the composition transformation that maps triangle \( \triangle LMN \) with vertices \( L(8, 3) \), \( M(4, 6) \), and \( N(5, 9) \) to triangle \( \triangle L'M'N' \) with vertices \( L'(-10, -3) \), \( M'(-6, 0) \), and \( N'(-7, 3) \), we can analyze the changes in the coordinates of each vertex.
Let's look at each vertex:
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For \( L(8, 3) \) to \( L'(-10, -3) \):
- Change in x-coordinate: \( -10 - 8 = -18 \)
- Change in y-coordinate: \( -3 - 3 = -6 \)
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For \( M(4, 6) \) to \( M'(-6, 0) \):
- Change in x-coordinate: \( -6 - 4 = -10 \)
- Change in y-coordinate: \( 0 - 6 = -6 \)
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For \( N(5, 9) \) to \( N'(-7, 3) \):
- Change in x-coordinate: \( -7 - 5 = -12 \)
- Change in y-coordinate: \( 3 - 9 = -6 \)
The y-coordinates for all transformations show a consistent decrease of 6 units down, indicating a downward movement.
Next, checking the transformation concerning the x-coordinates:
- The changes in the x-coordinates do not correspond to a simple horizontal translation.
- Observing the signs, if we reflect over the y-axis, we could see that reflecting the points \( (x, y) \) across the y-axis results in \( (-x, y) \).
- After reflecting, we will need to apply some translation to achieve the final coordinates.
Thus, we can determine the necessary reflection and translation:
- Reflection over the y-axis: This changes the x-coordinates from positive to negative.
- Translation: After reflecting, we will translate to the left by 6 and down by 6.
Therefore, the correct transformation rule is:
a reflection over the y-axis and a move to the left by 6 and down by 6.