A 2-dimensional shape was reflected over the x-axis, then rotated 270 degrees counterclockwise. What is true about the transformation?
Understand the Problem
The question is asking about the properties of a transformation involving a reflection and a rotation of a 2-dimensional shape. It seeks to determine whether this transformation preserves distance or not.
Answer
The transformation does preserve distance because the shape maintained the same size and only moved location.
Answer for screen readers
The transformation does preserve distance because the shape maintained the same size and only moved location.
Steps to Solve
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Identify the transformations The transformations involved are a reflection over the x-axis and a 270-degree counterclockwise rotation. We will analyze each transformation for distance preservation.
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Analyze the reflection When a shape is reflected over the x-axis, the x-coordinates of its points change signs, while the y-coordinates remain unchanged. This transformation preserves distances between all points because it is an isometric transformation.
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Analyze the rotation A rotation, such as rotating 270 degrees counterclockwise, does not change the size or shape of the figure. It also preserves the distances between all points because it is another type of isometric transformation.
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Combine the effects of the transformations Since both the reflection and the rotation preserve distances, the overall transformation of reflecting and then rotating also preserves distance.
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Evaluate the answer choices Now we can check which statement aligns with our analysis about distance preservation:
- The first choice mentions that distance is not preserved due to rotation (incorrect).
- The second choice focuses on the y-axis reflection (incorrect).
- The third choice wrongly states distances are preserved due to size increase (incorrect).
- The fourth choice correctly indicates distance preservation because the shape maintained the same size and only moved location (correct).
The transformation does preserve distance because the shape maintained the same size and only moved location.
More Information
Transformations such as reflections and rotations are examples of isometries, which are transformations that preserve distances. This means that the shape's dimensions remain unchanged during such transformations, although their positions may vary.
Tips
- Mixing up isometric transformations with those that do not preserve distance (like scaling) can lead to incorrect conclusions. Always verify the classification of the transformation.
- Misunderstanding the order of operations; reflect first, then rotate, ensuring each preserves distances before concluding about the overall transformation.
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