To analyze the composition of the transformations involving reflections, let's first break down what happens when triangle ABC is reflected over the line \( x = 2 \) and then reflected again over the line \( y = -2 \).
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First Reflection over the line \( x = 2 \): When a point \( (x, y) \) is reflected over the line \( x = 2 \), the new coordinates \( (x', y') \) become: \[ x' = 2 + (2 - x) = 4 - x \] The \( y \)-coordinate remains unchanged, so: \[ y' = y \] Therefore, any point \( (x, y) \) transforms to \( (4 - x, y) \).
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Second Reflection over the line \( y = -2 \): Next, when reflecting a point \( (x', y') \) over the line \( y = -2 \), the transformation is given by: \[ x'' = x' \quad (\text{remains unchanged}) \] \[ y'' = -2 + (-2 - y') = -4 - y' \] Since \( y' = y \) from the previous transformation, we get: \[ y'' = -4 - y \] Therefore, the point \( (x', y') \) transforms to \( (x', -4 - y) \).
So now, substituting our expression for \( x' \) gives the combined transformation of the initial point \( (x, y) \): \[ (x'', y'') = (4 - x, -4 - y) \]
Now, let's analyze how these transformations can be described systematically.
Both reflections across vertical and horizontal lines do not create a dilation but create a transformation resulting from reflections. In general, when you reflect about two perpendicular lines, the result is a rotation. In this case, the reflections over \( x = 2 \) and then \( y = -2 \) can be viewed as a rotation about a point that is located at the intersection of these lines, which is point \( (2, -2) \).
Thus, the resulting transformation can be described as:
B. a rotation about the intersection of the two perpendicular lines.
So the appropriate answer is B.
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