To identify the transformations that do not preserve distance but preserve angle measure, let's analyze each option:
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Reflection across the line \( y = x \): This transformation preserves both distance and angle measure, so it does not meet the criteria.
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Translation of 3 units upwards: Translations preserve both distance and angle measure, so it does not meet the criteria.
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Counterclockwise rotation of \( 180^\circ \) about the origin: Like translations and reflections, rotations preserve both distance and angle measure, so it does not meet the criteria.
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Dilation by a scale factor of 0.5 about the origin: Dilations do not preserve distance (since distances are scaled), but they preserve angle measures. Therefore, this transformation does meet the criteria.
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Dilation by a scale factor of -1 about the point (1, 1): This is a dilation that reflects points about the center and scales distances (with possible inversion). It does not preserve distance while preserving angle measures, fulfilling the criteria.
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Translation of 2 units to the right, then dilation by a scale factor of 1.5 about the point (1, 1): Here, the initial translation preserves both distance and angle measure. However, when followed by a dilation, which does not preserve distance, the final transformation does not preserve distance but preserves angle measures. Thus, it fulfills the criteria as well.
Final transformations that do not preserve distance but preserve angle measure:
- Dilation by a scale factor of 0.5 about the origin
- Dilation by a scale factor of -1 about the point (1, 1)
- Translation of 2 units to the right then dilation by a scale factor of 1.5 about the point (1, 1)