# Two opposite sides of a parallelogram each have a length of #14 #. If one corner of the parallelogram has an angle of #(3 pi)/4 # and the parallelogram's area is #70 #, how long are the other two sides?

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The length of the other two sides of the parallelogram can be found using the given information about the area and the angle.

Let ( a ) be the length of one of the adjacent sides to the given angle and ( b ) be the length of the other adjacent side.

Given that the area of the parallelogram is ( 70 ), and the angle ( \theta = \frac{3\pi}{4} ), we can use the formula for the area of a parallelogram:

[ \text{Area} = ab \sin(\theta) ]

Substituting the given values, we have:

[ 70 = 14b \sin\left(\frac{3\pi}{4}\right) ]

Solving for ( b ), we get:

[ b = \frac{70}{14 \sin\left(\frac{3\pi}{4}\right)} ]

Since ( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} ), we have:

[ b = \frac{70}{14 \cdot \frac{\sqrt{2}}{2}} = \frac{70}{7\sqrt{2}} = \frac{10}{\sqrt{2}} ]

Now, since the opposite sides of the parallelogram are equal, the length of the other two sides is also ( 14 ). Thus, the lengths of the other two sides are ( 14 ) each.

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