A parallelogram has sides A, B, C, and D. Sides A and B have a length of #2 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(3 pi)/4 #, what is the area of the parallelogram?

To find the area of the parallelogram, you can use the formula:

[ \text{Area} = \text{base} \times \text{height} ]

In a parallelogram, any side can be considered the base, and the height is the perpendicular distance between the base and the opposite side.

Given that sides A and C form an angle of ( \frac{3\pi}{4} ), we can find the height using trigonometry.

The height of the parallelogram can be calculated as follows:

[ \text{height} = \text{side } A \times \sin(\text{angle between A and C}) ]

Given that side A has a length of 2 and the angle between A and C is ( \frac{3\pi}{4} ):

[ \text{height} = 2 \times \sin\left(\frac{3\pi}{4}\right) ]

[ \text{height} = 2 \times \frac{\sqrt{2}}{2} ]

[ \text{height} = \sqrt{2} ]

Now, using the formula for the area of a parallelogram:

[ \text{Area} = \text{base} \times \text{height} ]

[ \text{Area} = 2 \times \sqrt{2} ]

[ \text{Area} = 2\sqrt{2} ]

So, the area of the parallelogram is ( 2\sqrt{2} ) square units.