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This question involves the dilation of a quadrilateral \( PQRS \) with the origin as the center of dilation to form quadrilateral \( P'Q'R'S' \).

To analyze the question:

1. **Dilation transformation**: Dilation involves scaling the figure by a certain factor with respect to the origin. Here, the figure is dilated, meaning its size is either increased or decreased while preserving the shape and angles.

2. **Answer choices**:
   - **A**: This option suggests that each side length of \( PQRS \) is half the corresponding side length of \( P'Q'R'S' \). This would indicate a dilation factor of 2 (since the side lengths of the original figure would need to be multiplied by 2 to get the larger figure).
   - **B**: This option suggests that \( P'Q'R'S' \) is congruent to \( PQRS \), meaning the two figures are identical in size and shape. This would only be true if the dilation factor were 1, which isn't indicated here.
   - **C**: This option suggests that each angle measure of \( PQRS \) is half the corresponding angle measure of \( P'Q'R'S' \). Dilation preserves angles, so this cannot be true.
   - **D**: This option suggests that \( P'Q'R'S' \) is similar to \( PQRS \). Since dilation transforms a figure into a similar figure by preserving angles and proportionality of sides, this statement should be true.

### Correct Answer:
**D**: Quadrilateral \( P'Q'R'S' \) is similar to quadrilateral \( PQRS \).

### Would you like more details, or do you have any questions?

### Related questions:
1. How do you calculate the scale factor in dilation?
2. What is the difference between congruence and similarity in geometry?
3. How does dilation affect angles and side lengths in geometric shapes?
4. How would a dilation factor greater than 1 affect the size of the image?
5. Can you explain why dilation preserves the shape of geometric figures?

### Tip:
In geometric transformations, dilation preserves the shape and angles but changes the size based on the scale factor.

Quadrilateral PQRS is dilated with the origin as the center of dilation to create quadrilateral P'Q'R'S'. The coordinates of each vertex are integers. Which statement is true?

Learn how to solve a geometry problem involving the dilation of quadrilateral PQRS to P'Q'R'S' with the origin as the center of dilation. This problem explores the concept of similarity and dilation in transformations. Suitable for Grades 9-10.

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