# Two opposite sides of a parallelogram each have a length of #9 #. If one corner of the parallelogram has an angle of #(7 pi)/8 # and the parallelogram's area is #18 #, how long are the other two sides?

The other two sides are 5.226 long each

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To find the lengths of the other two sides of the parallelogram, given that two opposite sides have a length of 9 and one angle is ( \frac{7\pi}{8} ), we can use the formula for the area of a parallelogram:

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

Given that the area is 18, we can set up the equation:

[ 18 = 9 \times \text{height} ]

Solving for the height, we find:

[ \text{height} = \frac{18}{9} = 2 ]

Now, the height of the parallelogram represents the length of the other two sides. Since the angle given is adjacent to one of the sides of length 9, we can use trigonometric ratios to find the length of the other sides.

Given that the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse, we can use:

[ \cos(\frac{7\pi}{8}) = \frac{\text{adjacent side}}{\text{hypotenuse}} ]

[ \cos(\frac{7\pi}{8}) = \frac{\text{height}}{\text{other side length}} ]

[ \text{other side length} = \frac{\text{height}}{\cos(\frac{7\pi}{8})} ]

Substituting the known values:

[ \text{other side length} = \frac{2}{\cos(\frac{7\pi}{8})} ]

We can use the fact that ( \cos(\frac{7\pi}{8}) = \cos(\frac{\pi}{8}) ) due to symmetry properties of the cosine function:

[ \cos(\frac{\pi}{8}) = \sqrt{\frac{1 + \cos(\frac{\pi}{4})}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} ]

Thus, the length of the other two sides of the parallelogram is:

[ \text{other side length} = \frac{2}{\sqrt{\frac{2 + \sqrt{2}}{4}}} = \frac{4}{\sqrt{2 + \sqrt{2}}} ]

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