Final answer:
The area of the parallelogram with given vertices is computed using the cross product of vectors representing adjacent sides AB and AD, which results in 16 square units.
Step-by-step explanation:
The question asks us to find the area of the parallelogram with vertices A(-4, 4), B(-2, 7), C(2, 5), and D(0, 2). To compute the area, we can use the formula for the area of a parallelogram given by the absolute value of the cross product of two adjacent sides. First, we need to find two vectors representing adjacent sides. Let's take vectors AB and AD:
- Vector AB = B - A = (-2 - (-4), 7 - 4) = (2, 3)
- Vector AD = D - A = (0 - (-4), 2 - 4) = (4, -2)
Now, the area of the parallelogram is given by:
|AB x AD| = |(2, 3) x (4, -2)|
The cross product of two-dimensional vectors AB and AD is a determinant:
|
2 4
3 -2
|
The magnitude of the cross product (which is the area in this two-dimensional case) is:
|AB x AD| = |(2 * -2) - (4 * 3)| = |-4 - 12| = | -16 | = 16
Thus, the area of the parallelogram is 16 square units.