Two opposite sides of a parallelogram each have a length of #8 #. If one corner of the parallelogram has an angle of #(5 pi)/6 # and the parallelogram's area is #44 #, how long are the other two sides?

To find the lengths of the other two sides of the parallelogram, we can use the formula for the area of a parallelogram and the given angle and length of one side.

Let's denote the length of one of the other two sides as ( a ) and the length of the adjacent side as ( b ).

The area of a parallelogram is given by the formula: ( \text{Area} = \text{base} \times \text{height} ). In this case, the base is one of the given sides with length 8, and the height is the length of the other side, which we denote as ( a ).

Thus, we have: ( 44 = 8a ).

This gives us the value of ( a ).

Next, we need to find the height of the parallelogram. The height can be calculated using trigonometry, as the side with length 8 is opposite to the given angle of ( \frac{5\pi}{6} ).

Using trigonometric ratios, we know that:

[ \sin\left(\frac{5\pi}{6}\right) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{\text{height}}{a} ]

Now we can solve for the height.

Once we find the height, we can use it to calculate the length of the other side ( b ) by subtracting twice the height from the given length of 8. This is because opposite sides of a parallelogram are equal in length.

So, the lengths of the other two sides of the parallelogram can be found using these calculations.