 
Explain how to prove that an image is the dilation of a preimage in the coordinate plane.

To prove that an image is the dilation of a preimage in the coordinate plane, you need to show that the image is a scaled version of the preimage, either enlarged or reduced, by a specific scale factor, and that all points are moved along lines through a fixed point called the center of dilation.

### Steps to Prove Dilation:

#### 1. **Identify the Center of Dilation**:
   - Determine the center of dilation, denoted as \( C \). This point is either on the coordinate plane or at the origin \((0,0)\). All points in the preimage are mapped along lines that pass through this center.

#### 2. **Corresponding Points**:
   - Identify corresponding points between the preimage and the image. If a point \( P(x, y) \) in the preimage corresponds to a point \( P'(x', y') \) in the image, these pairs should satisfy the dilation relationship.

#### 3. **Use the Dilation Formula**:
   - The relationship between the coordinates of corresponding points in a dilation is given by:
     \[
     P'(x', y') = \left( kx, ky \right)
     \]
     where \( k \) is the scale factor of the dilation.

   - If \( k > 1 \), the image is an enlargement of the preimage.
   - If \( 0 < k < 1 \), the image is a reduction of the preimage.

#### 4. **Check the Scale Factor**:
   - Calculate the ratio of the coordinates of each corresponding point \( P'(x', y') \) to \( P(x, y) \):
     \[
     k = \frac{x'}{x} = \frac{y'}{y}
     \]
   - If the ratios \( \frac{x'}{x} \) and \( \frac{y'}{y} \) are equal for all corresponding points and equal to the same constant \( k \), then the image is a dilation of the preimage by the scale factor \( k \).

#### 5. **Lines Through the Center of Dilation**:
   - Ensure that the line segment from the center of dilation \( C \) to any point on the preimage is parallel to the line segment from \( C \) to the corresponding point on the image. This confirms that the points have been scaled along lines radiating from the center.

#### 6. **Conclusion**:
   - If all the conditions above are met—i.e., each point on the image is a multiple of the corresponding point on the preimage by the same scale factor \( k \), and the lines through the center of dilation are consistent—you have proven that the image is a dilation of the preimage.

### Example:

Consider a triangle with vertices \( A(1, 2) \), \( B(3, 4) \), \( C(5, 6) \), and its dilated image \( A'(2, 4) \), \( B'(6, 8) \), \( C'(10, 12) \). If the center of dilation is the origin \((0,0)\), check the scale factor:

- For point \( A \): \( k = \frac{x'}{x} = \frac{2}{1} = 2 \) and \( k = \frac{y'}{y} = \frac{4}{2} = 2 \)
- For point \( B \): \( k = \frac{x'}{x} = \frac{6}{3} = 2 \) and \( k = \frac{y'}{y} = \frac{8}{4} = 2 \)
- For point \( C \): \( k = \frac{x'}{x} = \frac{10}{5} = 2 \) and \( k = \frac{y'}{y} = \frac{12}{6} = 2 \)

Since the scale factor \( k = 2 \) is consistent for all points, the image is a dilation of the preimage by a factor of 2.

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Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How does the location of the center of dilation affect the image?
2. What happens when the scale factor \( k \) is negative?
3. Can dilation change the orientation of a figure? If so, how?
4. How do you determine the center of dilation if it's not given?
5. What is the difference between dilation and other types of transformations like translation and rotation?

**Tip:** Always verify that the scale factor is consistent across all points to confirm a true dilation. Any discrepancy suggests a different transformation.

Learn how to prove that an image is the dilation of a preimage in the coordinate plane. Understand the dilation formula and its application in scaling figures. Suitable for students in Grades 6-8.

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