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The question asks for the coordinates of the image of point \( R(3, -2) \) after a dilation with center \( (0, 0) \) and a scale factor of \( \frac{1}{3} \).

### Solution:
The formula for dilation with a center at the origin \( (0, 0) \) is given by:
\[
(x', y') = (kx, ky)
\]
where \( (x, y) \) are the original coordinates, \( k \) is the scale factor, and \( (x', y') \) are the coordinates after dilation.

Given:
- Original point \( R(3, -2) \)
- Scale factor \( k = \frac{1}{3} \)

We apply the formula:
\[
x' = \frac{1}{3} \times 3 = 1
\]
\[
y' = \frac{1}{3} \times (-2) = -\frac{2}{3}
\]

So, the coordinates of the image of \( R \) after dilation are:
\[
R'(1, -\frac{2}{3})
\]

Would you like to go over more details on this, or have any further questions?

Here are five questions for further practice:
1. What are the coordinates of \( P(6, -1) \) after the same dilation?
2. How would you calculate the dilation of point \( Q(3, 4) \) with a different scale factor, say \( 2 \)?
3. How does the distance between the points change after dilation by \( \frac{1}{3} \)?
4. What would happen to point \( R \) if the scale factor was negative, say \( -\frac{1}{3} \)?
5. Can dilation be performed with a center point that is not \( (0, 0) \)? How would the formula change?

**Tip**: When performing dilation, always multiply both the x and y coordinates by the scale factor to get the new location.

What are the coordinates of the image of R for a dilation with center (0, 0) and scale factor 1/3?

Learn how to find the coordinates of point R after a dilation with a scale factor of 1/3 and a center at the origin (0, 0). This step-by-step guide covers the key formula and transformation concepts, suitable for students in Grades 8-10.

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