# A parallelogram has sides A, B, C, and D. Sides A and B have a length of #5 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(5 pi)/8 #, what is the area of the parallelogram?

So let's sketch the scenario. Remember that in a parallelogram, opposite angles are congruent and adjacent angles are supplementary.

To find the area, we will split the shape into three shapes. Two right angled triangles and a rectangle.

We find the length of the vertical lines (the height of the triangles) by trigonometry. Remember SOH CAH TOA, so sine is opposite/hypotenuse.

We find the base of the triangles with the same technique:

The area of each triangle is given by

If we use

The area outside the rectangle is given by

The rectangle's dimensions are given by the height of the triangles and

As a sense check, if the shape was just a rectangle the area would be 40 so this is in the right neck of the woods at least.

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To find the area of the parallelogram, you can use the formula:

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

where the base is the length of either side A or B, and the height is the perpendicular distance between sides A and C.

Given that the length of side A (and also side B) is 5, and the length of sides C (and also side D) is 8, we can use the formula:

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

Solving this, the area of the parallelogram is approximately 22.57 square units.

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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 with lengths of #15 # and #12 #. If the parallelogram's area is #66 #, what is the length of its longest diagonal?
- A parallelogram has sides with lengths of #16 # and #5 #. If the parallelogram's area is #48 #, what is the length of its longest diagonal?
- Two opposite sides of a parallelogram have lengths of #7 #. If one corner of the parallelogram has an angle of #pi/4 # and the parallelogram's area is #42 #, how long are the other two sides?
- A parallelogram has sides with lengths of #21 # and #12 #. If the parallelogram's area is #84 #, what is the length of its longest diagonal?
- A rectangle has an area of 12 square centimeters and a perimeter of 16 centimeters. Which of the following could be its dimensions?

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