Two opposite sides of a parallelogram each have a length of #8 #. If one corner of the parallelogram has an angle of #(11 pi)/12 # and the parallelogram's area is #72 #, how long are the other two sides?

To find the lengths of the other two sides of the parallelogram, you can use the formula for the area of a parallelogram, which is given by the product of the length of one side and the perpendicular distance from that side to the opposite side.

Given that one pair of opposite sides each have a length of 8 and the area of the parallelogram is 72, you can set up the equation (8h = 72), where (h) represents the height of the parallelogram.

Once you find the height, you can use the given angle to determine the length of the other two sides using trigonometric functions.

Given that the opposite sides of the parallelogram have a length of 8 and the area is 72, we can use the formula for the area of a parallelogram, which is the product of the base and the height.

Let's denote one of the unknown sides as ( a ) and the other as ( b ). The height of the parallelogram can be found by using the formula ( h = ab \sin(\theta) ), where ( \theta ) is the angle between the two known sides.

Given that the area is 72 and one of the known sides is 8, we can find ( h ) using the formula:

[ 72 = 8a \sin\left(\frac{11\pi}{12}\right) ]

Solving for ( a ):

[ a = \frac{72}{8 \sin\left(\frac{11\pi}{12}\right)} ]

[ a ≈ \frac{72}{8 \sin\left(\frac{11\pi}{12}\right)} ]

Once we find the value of ( a ), we can find ( b ) using the formula for the area:

[ 72 = a \cdot b \cdot \sin\left(\frac{\pi}{12}\right) ]

[ b = \frac{72}{a \sin\left(\frac{\pi}{12}\right)} ]

Then, substitute the value of ( a ) into the equation to find ( b ).