# A parallelogram has sides A, B, C, and D. Sides A and B have a length of #4 # and sides C and D have a length of # 5 #. If the angle between sides A and C is #(5 pi)/6 #, what is the area of the parallelogram?

Always a good idea to draw a diagram. It helps to see what is going on!

You can observe that if vertical lines are dropped from each end of D we have a rectangle. So the area can be determined by

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Observe that

Thus

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Thus area

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To find the area of the parallelogram, we use the formula:

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

Given that sides A and B have a length of 4 and sides C and D have a length of 5, the base of the parallelogram is either A or C. Since we know the angle between sides A and C, we can use trigonometry to find the height of the parallelogram.

The height (( h )) can be found using the formula:

[ h = |AB| \times \sin(\theta) ]

where ( |AB| ) is the length of side A (or side C) and ( \theta ) is the angle between sides A and C.

Substituting the given values:

[ h = 4 \times \sin\left(\frac{5\pi}{6}\right) ]

[ h = 4 \times \sin\left(150^\circ\right) ]

[ h = 4 \times \left(-\frac{\sqrt{3}}{2}\right) ]

[ h = -2\sqrt{3} ]

Since the height cannot be negative in this context, we take the absolute value.

[ h = 2\sqrt{3} ]

Now, we have the base (4) and the height (2√3). Substituting these values into the formula for the area of a parallelogram:

[ \text{Area} = 4 \times 2\sqrt{3} ]

[ \text{Area} = 8\sqrt{3} ]

Therefore, the area of the parallelogram is ( 8\sqrt{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.

- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #9 # and sides C and D have a length of # 4 #. If the angle between sides A and C is #(5 pi)/12 #, what is the area of the parallelogram?
- Two opposite sides of a parallelogram each have a length of #15 #. If one corner of the parallelogram has an angle of #(5 pi)/12 # and the parallelogram's area is #160 #, how long are the other two sides?
- A parallelogram has sides with lengths of #15 # and #8 #. If the parallelogram's area is #15 #, what is the length of its longest diagonal?
- Two opposite sides of a parallelogram have lengths of #12 #. If one corner of the parallelogram has an angle of #pi/8 # and the parallelogram's area is #32 #, how long are the other two sides?
- A parallelogram has sides with lengths of #12 # and #4 #. If the parallelogram's area is #12 #, what is the length of its longest diagonal?

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