To analyze the composition of the transformations given in the problem, we first break down the steps:
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Reflection over the line \( x = 2 \): When a point \( (x, y) \) is reflected over the vertical line \( x = 2 \), the new coordinates of the reflected point will be \( (4 - x, y) \).
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Reflection over the line \( y = -2 \): When a point \( (x, y) \) is reflected over the horizontal line \( y = -2 \), the new coordinates of the reflected point will be \( (x, -4 - y) \).
To analyze the overall transformation, let's consider a point \( P(x, y) \):
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After the first reflection over \( x = 2 \): \[ P' = (4 - x, y) \]
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Then, reflecting \( P' \) over \( y = -2 \): \[ P'' = (4 - x, -4 - y) \]
We can analyze how this transformation changes the coordinates.
Now, let's define:
- \( x' = 4 - x \)
- \( y' = -4 - y \)
From here, we can express the final point \( P'' \) in terms of the original point \( P \):
- \( P'' = (4 - x, -4 - y) \)
To find a single transformation that summarizes these two reflections:
- The first transformation changes \( x \) to \( 4 - x \).
- The second transformation changes \( y \) to \( -4 - y \).
Ultimately, this can be thought of as a rotation. Specifically, a composition of two reflections across two orthogonal lines results in a rotation about the intersection point of those two lines.
Conclusion:
The combination of reflections across two perpendicular lines results in a rotation. Therefore, the best description of the composition transformation using a single transformation is:
a rotation about the intersection of the two perpendicular lines.
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