https://file.aiquickdraw.com/mathbot/file/1729454169273dyy5ntkc.jpg

From the image, I can deduce the transformation involves a translation and a dilation of triangle \( \triangle XYZ \) to \( \triangle X'Y'Z' \). The task is to find the scale factor of the dilation by comparing the sizes of corresponding sides of \( \triangle XYZ \) and \( \triangle X'Y'Z' \).

### Steps to find the scale factor:
1. **Locate a corresponding pair of sides**: For example, the side between \( X \) and \( Y \) in \( \triangle XYZ \) and between \( X' \) and \( Y' \) in \( \triangle X'Y'Z' \).
2. **Measure or calculate the lengths of these sides**.
   
   - The side \( XY \) (on the larger blue triangle) has a length of approximately 15 units (from point \( X \) at \( (-10, 0) \) to point \( Y \) at \( (5, 0) \)).
   - The side \( X'Y' \) (on the smaller yellow triangle) has a length of 3 units (from point \( X' \) at \( (-5, 6) \) to point \( Y' \) at \( (0, 6) \)).
   
3. **Calculate the scale factor**:
   \[
   \text{Scale factor} = \frac{\text{Length of } X'Y'}{\text{Length of } XY} = \frac{3}{15} = \frac{1}{5}
   \]

Thus, the scale factor for the dilation is \( \frac{1}{5} \).

### Summary:
- The scale factor of the dilation is \( \frac{1}{5} \).

Would you like further details on how the translation works, or do you have any questions?

Here are 5 related questions to expand on this concept:
1. What is the general formula for finding the scale factor in a dilation?
2. How does translation affect the coordinates of points in a triangle?
3. Can a dilation have a scale factor larger than 1? What does that imply?
4. How would you reverse a dilation with a scale factor of \( \frac{1}{5} \)?
5. How is the center of dilation determined in these transformations?

**Tip**: When dealing with dilations, always ensure to compare corresponding sides or distances from the center of dilation to maintain accuracy.

Learn how to calculate the scale factor and the translation rule for transforming a triangle with a dilation centered at the origin. This problem involves calculating the side lengths and applying the scale factor formula. Suitable for Grades 8-10.

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