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To solve this problem, let's go through each part systematically.

### Part (a): Determine the coordinates of \( R' \), \( S' \), and \( T' \)

A dilation transformation with a scale factor of \( \frac{3}{2} \) means that each coordinate \( (x, y) \) will be scaled by multiplying both \( x \) and \( y \) by \( \frac{3}{2} \).

Let's apply this to each point:

1. **For \( R(6, 4) \):**
   \[
   R' = \left(\frac{3}{2} \times 6, \frac{3}{2} \times 4\right) = (9, 6)
   \]

2. **For \( S(6, -2) \):**
   \[
   S' = \left(\frac{3}{2} \times 6, \frac{3}{2} \times -2\right) = (9, -3)
   \]

3. **For \( T(-2, -4) \):**
   \[
   T' = \left(\frac{3}{2} \times -2, \frac{3}{2} \times -4\right) = (-3, -6)
   \]

So the answers for part (a) are:
- \( R' = (9, 6) \)
- \( S' = (9, -3) \)
- \( T' = (-3, -6) \)

### Part (b): Choose the correct rule for the dilation mapping

The dilation rule with a scale factor of \( \frac{3}{2} \) applied to any point \( (x, y) \) is:
\[
(x, y) \rightarrow \left(\frac{3}{2}x, \frac{3}{2}y\right)
\]

Looking at the options provided, the correct answer for part (b) is:
- **\( (x, y) \rightarrow \left(\frac{3}{2}x, \frac{3}{2}y\right) \)**.

### Summary

1. **Coordinates after dilation:**
   - \( R' = (9, 6) \)
   - \( S' = (9, -3) \)
   - \( T' = (-3, -6) \)

2. **Dilation rule:** \( (x, y) \rightarrow \left(\frac{3}{2}x, \frac{3}{2}y\right) \)

Would you like further details on dilation transformations?

### Related Questions

1. How would the coordinates change if the scale factor was \( \frac{1}{2} \) instead?
2. What is the effect of a scale factor less than 1 on the shape of the triangle?
3. How do you find the area of \( \triangle R'S'T' \) after dilation?
4. What happens to a shape if the scale factor is negative?
5. How does a dilation differ from a rotation in terms of coordinate changes?

### Tip
Remember, in a dilation centered at the origin, each coordinate is simply scaled by the factor, making calculations straightforward.

A dilation centered at the origin with a scale factor of 3/2 is applied to triangle RST. The result is triangle R'S'T'.

Understand how to perform a dilation transformation on triangle RST with a scale factor of 3/2. Learn to find the coordinates of the transformed triangle R'S'T' and apply the dilation formula for accurate results. Suitable for students in Grades 9-10.

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