# Two opposite sides of a parallelogram each have a length of #6 #. If one corner of the parallelogram has an angle of #(5 pi)/12 # and the parallelogram's area is #54 #, how long are the other two sides?

Other two sides are

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To find the lengths of the other two sides of the parallelogram, we can use the given information about its area, the lengths of the opposite sides, and the angle at one of its corners.

The formula to find the area of a parallelogram is: Area = base × height.

Given that the area is 54 and one of the sides (which is also the base) has a length of 6, we can find the height of the parallelogram.

Area = 54 Base = 6

Using the formula, we can find the height:

Height = Area / Base = 54 / 6 = 9

Now, we need to find the lengths of the other two sides. Since the opposite sides of a parallelogram are equal in length, the length of the other side opposite to the side with length 6 is also 6.

To find the lengths of the remaining sides, we will use trigonometric ratios, specifically the sine and cosine functions, based on the given angle.

Given angle: (5π)/12

The cosine of the given angle will give us the ratio of the adjacent side to the hypotenuse, and the sine will give us the ratio of the opposite side to the hypotenuse. Since we already know the length of the opposite side (height), we can use the sine function to find the length of the adjacent side.

Cos((5π)/12) = adjacent / 6

Since Cos((5π)/12) ≈ 0.258819, we have:

0.258819 = adjacent / 6

Adjacent = 6 × 0.258819 ≈ 1.553

So, one of the other sides has a length of approximately 1.553.

As the opposite sides of a parallelogram are equal, the length of the remaining side is also 1.553.

Therefore, the lengths of the other two sides are approximately 1.553 each.

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