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.