To find the area of the parallelogram, we need to use the coordinates of its vertices.

The vertices of the parallelogram are located at:

• (4, 0)
• (10, 4)
• (12, 0)
• (6, -4)

Method: Using the Determinant Formula

We can use the determinant formula to find the area of a polygon given its vertices: $\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|$

Let: $(x_1, y_1) = (4, 0), \, (x_2, y_2) = (10, 4), \, (x_3, y_3) = (12, 0), \, (x_4, y_4) = (6, -4)$

Plugging these values into the formula:

$\text{Area} = \frac{1}{2} \left| (4 \cdot 4 + 10 \cdot 0 + 12 \cdot -4 + 6 \cdot 0) - (0 \cdot 10 + 4 \cdot 12 + 0 \cdot 6 + -4 \cdot 4) \right|$

Let's compute this step-by-step.The area of the parallelogram is 32 square units.

Would you like more details on how this calculation was performed, or do you have any other questions?

Here are some related questions to explore:

1. How can you find the area of a triangle using the determinant method?
2. What other methods can be used to find the area of a parallelogram on a coordinate plane?
3. How can the coordinates of a point be used to determine if it lies inside a polygon?
4. What is the formula for the area of a parallelogram using base and height?
5. How can the distance formula be used to calculate the lengths of sides in coordinate geometry?

Tip: When dealing with polygons on a coordinate plane, using the determinant formula is an efficient way to calculate areas, especially for irregular shapes.