To find the coordinates of vertex $D'$ after a dilation with respect to the origin by a scale factor of $k = 12$, we can use the dilation formula:

$(x', y') = (k \cdot x, k \cdot y)$

For vertex $D$, with coordinates $(2, -2)$:

1. Apply the dilation by multiplying each coordinate of $D$ by the scale factor $12$: $x' = 12 \cdot 2 = 24$ $y' = 12 \cdot (-2) = -24$

2. So, the coordinates of $D'$ are: $D' = (24, -24)$

Would you like further details on dilation or other points in the shape?

Further Questions:

1. How would the coordinates of point $A$ change under the same dilation?
2. What are the new coordinates of point $B$ after applying a dilation of 12?
3. What happens to the area of the parallelogram after a dilation with this factor?
4. What would be the coordinates of $C'$ after a dilation of 12?
5. How does a dilation centered at a different point affect the coordinates of each vertex?

Tip: When dilating points with respect to the origin, simply multiply each coordinate by the scale factor!