# Two opposite sides of a parallelogram each have a length of #12 #. If one corner of the parallelogram has an angle of #(3pi)/8 # and the parallelogram's area is #72 #, how long are the other two sides?

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Given that the opposite sides of the parallelogram have a length of 12 and the area of the parallelogram is 72, we can use the formula for the area of a parallelogram:

[ \text{Area} = \text{base} \times \text{height} ]

In a parallelogram, the base is one of the sides, and the height is the perpendicular distance between the side and its opposite side.

Let's denote one of the other two sides as ( a ) and the height (perpendicular distance) from that side to the base as ( h ).

Given that the area is 72 and one of the sides (base) is 12, we can find the height ( h ) using the formula:

[ 72 = 12 \times h ]

[ h = \frac{72}{12} = 6 ]

Now, we can use the height and the given angle to find the length of the other two sides using trigonometry. Since the given angle is ( \frac{3\pi}{8} ), and it's opposite to one of the sides of length 12, we can use the cosine function to find the length of the adjacent side ( a ):

[ \cos\left(\frac{3\pi}{8}\right) = \frac{\text{adjacent side}}{\text{hypotenuse}} ]

[ \text{adjacent side} = 12 \cdot \cos\left(\frac{3\pi}{8}\right) ]

[ \text{adjacent side} = 12 \cdot \cos\left(\frac{3\pi}{8}\right) ]

[ \text{adjacent side} \approx 12 \cdot 0.3827 \approx 4.592 ]

So, the lengths of the other two sides are approximately 4.592 units each.

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The length of the other two sides of the parallelogram is 6.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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