# Two opposite sides of a parallelogram each have a length of #12 #. If one corner of the parallelogram has an angle of #(5 pi)/6 # and the parallelogram's area is #32 #, how long are the other two sides?

The other two sides are 16 long.

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The area of a parallelogram can be calculated using the formula:

Area = base × height

In this case, the area of the parallelogram is given as 32 square units. Given that one corner of the parallelogram has an angle of (5π)/6, the height of the parallelogram can be determined using trigonometry.

Using the formula for the area of a parallelogram, we can rearrange it to solve for the height:

Height = Area / Base

Substitute the given values:

Height = 32 / 12

Calculate the height:

Height = 2.6667

Now, using trigonometry, we can find the length of the other two sides (let's call them A and B) by using the given angle (5π)/6:

A = B × sin(angle)

Substitute the given values:

A = 12 × sin(5π/6)

Calculate the sine of the angle:

sin(5π/6) = √3/2

Now, calculate the length of side A:

A = 12 × √3/2 ≈ 10.3923

Since the opposite sides of a parallelogram are equal in length, the length of side B is also approximately 10.3923 units.

So, the lengths of the other two sides of the parallelogram are approximately 10.3923 units each.

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