# 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 #72 #, how long are the other two sides?

Other two sides are

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To find the lengths of the other two sides of the parallelogram, we can use the formula for the area of a parallelogram, which is given by the product of the base and the height. In this case, the base will be one of the sides with length 8, and the height will be the length of the perpendicular from one side to the opposite side.

Given that the area of the parallelogram is 72, and one side has a length of 8, we can rearrange the formula for the area to solve for the height. Thus, the height of the parallelogram is given by:

[ \text{Area} = \text{Base} \times \text{Height} ] [ 72 = 8 \times \text{Height} ]

[ \text{Height} = \frac{72}{8} = 9 ]

Now, we know the height of the parallelogram is 9. The length of the perpendicular is the distance from the given angle to the opposite side. Since the given angle is (5π)/6, which is 150 degrees in degrees, the angle between the given side and the height is the complement of 150 degrees, which is 30 degrees.

Using trigonometric ratios, we can find the length of the perpendicular:

[ \text{Opposite side} = \text{Adjacent side} \times \tan(\text{Angle}) ] [ \text{Opposite side} = 8 \times \tan(30^\circ) ]

[ \text{Opposite side} = 8 \times \frac{\sqrt{3}}{3} = \frac{8\sqrt{3}}{3} ]

Therefore, both opposite sides of the parallelogram are ( \frac{8\sqrt{3}}{3} ).

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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.

- Two opposite sides of a parallelogram each have a length of #7 #. If one corner of the parallelogram has an angle of #( pi)/3 # and the parallelogram's area is #42 #, how long are the other two sides?
- Two rhombuses have sides with lengths of #15 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #pi/2 #, what is the difference between the areas of the rhombuses?
- Two opposite sides of a parallelogram each have a length of #36 #. If one corner of the parallelogram has an angle of #(5 pi)/12 # and the parallelogram's area is #64 #, how long are the other two sides?
- Two opposite sides of a parallelogram each have a length of #6 #. If one corner of the parallelogram has an angle of #(7 pi)/8 # and the parallelogram's area is #18 #, how long are the other two sides?
- What is a quadrilateral that has exactly 2 congruent consecutive sides? Would it be either a parallelogram and/or a trapezoid?

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