The area of the parallelogram with vertices $(1,2)$, $(6,3)$, $(7,8)$, and $(12,9)$ is $24$ square units.

Would you like details on the solution or have any questions? Here are five related questions to expand your understanding:

1. How do you find the area of a parallelogram using the coordinates of its vertices?
2. What is the significance of the cross product in calculating the area of a parallelogram?
3. How can you verify that the given points form a parallelogram?
4. How does the formula for the area of a parallelogram differ from that of a triangle?
5. What are other methods to calculate the area of a parallelogram?

Tip: For any set of points, plotting them on a graph can help visualize the problem and understand the geometric relationships between the points.