To find the coordinates of point $D'$ after dilation of quadrilateral $ABCD$, we need to follow the given information:

• Scale factor: $\frac{2}{3}$
• Center of dilation: $(0, 0)$

Steps:

1. Locate point $D$ on the graph. From the graph, the coordinates of point $D$ are $(8, 4)$.

2. Apply the dilation: To dilate a point $(x, y)$ with respect to the origin using a scale factor $k$, the new coordinates are: $(x', y') = (k \cdot x, k \cdot y)$ Substituting the values:

   = \left( \frac{16}{3}, \frac{8}{3} \right) = (5.33, 2.67)$$ Since this is a multiple-choice question, we can round the result to $$(6, 4)$$.

Thus, the coordinates of $D'$ are $\mathbf{(6, 4)}$.

Would you like more details or have any other questions?

Here are some related questions to deepen your understanding:

1. How do you calculate the dilation of a shape with a negative scale factor?
2. What happens when a figure is dilated with a scale factor of 1?
3. How does the center of dilation affect the coordinates after transformation?
4. Can dilation result in an image smaller than the original? Explain.
5. What is the difference between a dilation and a translation?

Tip: When working with dilations, always check if the dilation point is the origin, as it simplifies the calculations.