Two opposite sides of a parallelogram each have a length of #2 #. If one corner of the parallelogram has an angle of #(7 pi)/8 # and the parallelogram's area is #5 #, how long are the other two sides?

sides and corner angle respectively.

The other two sides of the parallelogram are each of length √5.

Given that two opposite sides of the parallelogram each have a length of 2 units and one corner of the parallelogram has an angle of ( \frac{7\pi}{8} ), and the area of the parallelogram is 5 square units, the length of the other two sides can be calculated as follows:

1. Find the height (h) of the parallelogram using the formula for the area of a parallelogram: [ \text{Area} = \text{base} \times \text{height} ]

2. The base is the side of length 2 units. So, the height can be calculated as: [ h = \frac{\text{Area}}{\text{base}} = \frac{5}{2} ]

3. Now, use trigonometric relationships in a right triangle formed by the height and one of the sides to find the length of the other two sides.

Given that the angle opposite the side of length 2 is ( \frac{7\pi}{8} ), we can use trigonometric functions to find the lengths of the other two sides.

4. Let's call one of the other two sides x. Using trigonometry, we can write: [ \cos\left(\frac{7\pi}{8}\right) = \frac{2}{x} ] [ x = \frac{2}{\cos\left(\frac{7\pi}{8}\right)} ]

5. Once we find the value of x, we can use the properties of a parallelogram to conclude that the length of the other remaining side is also x.

Using the above calculations, we can determine the lengths of the other two sides of the parallelogram.