https://file.aiquickdraw.com/mathbot/file/17273182846697ev65p0x.jpg

The problem states that the given figure is dilated by a factor of \( \frac{4}{3} \) centered at the origin. To help you plot the resulting image, let’s work through the dilation process step by step:

### Steps to Dilate the Figure:

1. **Identify the coordinates of the vertices** of the original triangle. From the image, the points seem to be approximately:
   - \( Q = (-6, 4) \)
   - \( P = (6, 4) \)
   - \( N = (-6, -8) \)

2. **Apply the dilation formula**:
   - Dilation about the origin by a factor of \( \frac{4}{3} \) means multiplying each coordinate of the points by \( \frac{4}{3} \).

   So, for each vertex \( (x, y) \), the new coordinates will be:
   \[
   (x', y') = \left( \frac{4}{3}x, \frac{4}{3}y \right)
   \]

   - For \( Q = (-6, 4) \):
     \[
     Q' = \left( \frac{4}{3} \times -6, \frac{4}{3} \times 4 \right) = (-8, \frac{16}{3}) \approx (-8, 5.33)
     \]
   
   - For \( P = (6, 4) \):
     \[
     P' = \left( \frac{4}{3} \times 6, \frac{4}{3} \times 4 \right) = (8, \frac{16}{3}) \approx (8, 5.33)
     \]

   - For \( N = (-6, -8) \):
     \[
     N' = \left( \frac{4}{3} \times -6, \frac{4}{3} \times -8 \right) = (-8, -\frac{32}{3}) \approx (-8, -10.67)
     \]

### The dilated coordinates:
- \( Q' = (-8, 5.33) \)
- \( P' = (8, 5.33) \)
- \( N' = (-8, -10.67) \)

These are the new positions of the vertices after dilation. You can now plot these points and connect them to form the dilated image.

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Would you like more details or help with plotting this?

Here are 5 follow-up questions:
1. What happens to the angles of a shape after dilation?
2. How does a dilation factor greater than 1 affect the size of the shape?
3. What would the dilation look like with a factor less than 1?
4. Can a dilation move the figure to another quadrant?
5. How do you find the center of dilation if it's not the origin?

**Tip**: When dilating by a factor \( k \), if \( k > 1 \), the figure enlarges; if \( 0 < k < 1 \), the figure shrinks.

The figure below is dilated by a factor of 4/3 centered at the origin. Plot the resulting image.

This problem explores the dilation of a triangle by a factor of 4/3 centered at the origin. The solution involves applying the dilation formula to transform the coordinates of the triangle's vertices, yielding the new positions after dilation. Suitable for students in Grades 8-10.

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