# Two opposite sides of a parallelogram each have a length of #5 #. If one corner of the parallelogram has an angle of #(5 pi)/8 # and the parallelogram's area is #12 #, how long are the other two sides?

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Given that two opposite sides of a parallelogram each have a length of 5, and one corner of the parallelogram has an angle of ( \frac{5\pi}{8} ), we can use the formula for the area of a parallelogram to find the lengths of the other two sides.

The area of a parallelogram is given by the formula ( A = ab \sin(\theta) ), where ( A ) is the area, ( a ) and ( b ) are the lengths of the sides, and ( \theta ) is the angle between the sides.

Given ( A = 12 ), ( a = b = 5 ), and ( \theta = \frac{5\pi}{8} ), we can solve for the lengths of the other two sides.

[ 12 = 5 \times 5 \times \sin\left(\frac{5\pi}{8}\right) ]

[ 12 = 25 \times \sin\left(\frac{5\pi}{8}\right) ]

[ \sin\left(\frac{5\pi}{8}\right) = \frac{12}{25} ]

Now, we can use the sine ratio to find the lengths of the other two sides. Let ( x ) be the length of one of these sides. Then, the length of the other side is also ( x ).

[ \sin\left(\frac{5\pi}{8}\right) = \frac{\text{opposite side}}{\text{hypotenuse}} ]

[ \sin\left(\frac{5\pi}{8}\right) = \frac{x}{5} ]

[ \frac{12}{25} = \frac{x}{5} ]

[ x = \frac{12}{25} \times 5 ]

[ x = \frac{60}{25} ]

[ x = 2.4 ]

So, the lengths of the other two sides are both ( 2.4 ).

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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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