https://file.aiquickdraw.com/mathbot/file/1727878121884sbjgbofv.jpg

To solve this problem, we need to find the scale factor used to dilate triangle \( \Delta ABC \) to triangle \( \Delta A'B'C' \).

### Step 1: Identify the coordinates before and after dilation
- Triangle \( \Delta ABC \) has vertices:
  - \( A(2, -4) \)
  - \( B(6, -2) \)
  - \( C(4, 8) \)
  
- After dilation, triangle \( \Delta A'B'C' \) has vertices:
  - \( A'(3, -6) \)
  - \( B'(9, -3) \)
  - \( C'(6, 12) \)

### Step 2: Use the formula for dilation
The formula for dilation is:
\[
(x', y') = (kx, ky)
\]
where \( k \) is the scale factor, and \( (x, y) \) are the original coordinates.

### Step 3: Calculate the scale factor for each point
- From \( A(2, -4) \) to \( A'(3, -6) \):
  \[
  k = \frac{3}{2} = \frac{-6}{-4} = 1.5
  \]
  
- From \( B(6, -2) \) to \( B'(9, -3) \):
  \[
  k = \frac{9}{6} = \frac{-3}{-2} = 1.5
  \]
  
- From \( C(4, 8) \) to \( C'(6, 12) \):
  \[
  k = \frac{6}{4} = \frac{12}{8} = 1.5
  \]

### Step 4: Conclusion
The scale factor used to dilate \( \Delta ABC \) to \( \Delta A'B'C' \) is \( \boxed{1.5} \).

Would you like further details or have any additional questions? 

Here are five related questions to expand the topic:
1. How would you determine the center of dilation?
2. What happens if the scale factor is less than 1?
3. Can you perform a dilation with a negative scale factor?
4. How do you find the area of a dilated triangle compared to the original?
5. How would the scale factor affect the distance between points?

**Tip:** When dilating, the scale factor applies equally to both the x and y coordinates, ensuring the proportional enlargement or reduction of the figure.

Triangle ABC has vertices A(2, -4), B(6, -2), and C(4, 8). After a dilation, the image has vertices A'(3, -6), B'(9, -3), and C'(6, 12). What scale factor was used to dilate ΔABC to ΔA'B'C'?

This problem involves finding the scale factor used to dilate triangle ABC with vertices A(2, -4), B(6, -2), and C(4, 8) to triangle A'B'C' with vertices A'(3, -6), B'(9, -3), and C'(6, 12). Learn how to calculate scale factors and understand geometric transformations using the dilation formula.

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