https://file.aiquickdraw.com/mathbot/file/1727442180155mvra7qbx.jpg

The figure in the image needs to be dilated by a factor of \( \frac{4}{3} \), centered at the origin. This means that every point in the figure will be scaled outward from the origin by this factor. To calculate the new coordinates of each point after dilation, multiply the coordinates of each point by \( \frac{4}{3} \).

Let's analyze the coordinates of the points \( U \), \( T \), \( R \), and \( S \) before the dilation:

- \( U(-6, 2) \)
- \( T(0, 1) \)
- \( R(-6, -4) \)
- \( S(4, -3) \)

Now, applying the dilation factor \( \frac{4}{3} \):

1. **Dilating \( U(-6, 2) \):**
   \[
   U' = \left( -6 \times \frac{4}{3}, 2 \times \frac{4}{3} \right) = (-8, \frac{8}{3}) \approx (-8, 2.67)
   \]

2. **Dilating \( T(0, 1) \):**
   \[
   T' = \left( 0 \times \frac{4}{3}, 1 \times \frac{4}{3} \right) = (0, \frac{4}{3}) \approx (0, 1.33)
   \]

3. **Dilating \( R(-6, -4) \):**
   \[
   R' = \left( -6 \times \frac{4}{3}, -4 \times \frac{4}{3} \right) = (-8, -\frac{16}{3}) \approx (-8, -5.33)
   \]

4. **Dilating \( S(4, -3) \):**
   \[
   S' = \left( 4 \times \frac{4}{3}, -3 \times \frac{4}{3} \right) = \left( \frac{16}{3}, -\frac{12}{3} \right) \approx (5.33, -4)
   \]

### New Coordinates after Dilation:
- \( U'(-8, 2.67) \)
- \( T'(0, 1.33) \)
- \( R'(-8, -5.33) \)
- \( S'(5.33, -4) \)

Would you like me to plot the resulting image with these new points, or explain the dilation process in further detail?

Here are 5 related questions:
1. What happens to the angles of a figure when dilated?
2. How does dilation affect the orientation of a figure?
3. How would the figure change if the dilation factor was less than 1?
4. Can you explain why dilation preserves the shape of a figure?
5. How is dilation different from other transformations like translation or reflection?

**Tip:** When dilating around the origin, every point on the figure moves along a straight line extending from the origin, with distances scaled by the dilation factor.

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

This problem involves dilating a geometric figure centered at the origin by a factor of 4/3. The solution demonstrates how to apply the dilation formula to each vertex of the figure to obtain the new coordinates, preserving the shape but altering the size. Suitable for students in Grades 8-10.

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