# A parallelogram has sides with lengths of #15 # and #12 #. If the parallelogram's area is #99 #, what is the length of its longest diagonal?

The length of the longest diagonal is

Let

and

Then,

the area of the parallegram is

Therefore,

Then,

the angle

We apply the cosine rule to the triangle

So,

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The length of the longest diagonal of the parallelogram can be found using the formula:

[ \text{Longest diagonal} = \sqrt{a^2 + b^2 + 2ab\cos(\theta)} ]

where (a) and (b) are the lengths of the sides of the parallelogram, and (\theta) is the angle between those sides. Given that the area of the parallelogram is 99 and the lengths of the sides are 15 and 12, we can find the angle between the sides using the formula for the area of a parallelogram:

[ \text{Area} = ab\sin(\theta) ]

Substituting the given values:

[ 99 = 15 \cdot 12 \cdot \sin(\theta) ]

From this, we can solve for (\sin(\theta)), then find (\cos(\theta)) since (\sin(\theta)^2 + \cos(\theta)^2 = 1).

Once we have (\cos(\theta)), we can substitute all the values into the formula for the longest diagonal to find its length.

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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 #2 #. If one corner of the parallelogram has an angle of #(2 pi)/3 # and the parallelogram's area is #15 #, how long are the other two sides?
- Two rhombuses have sides with lengths of #1 #. If one rhombus has a corner with an angle of #pi/6 # and the other has a corner with an angle of #(3pi)/4 #, what is the difference between the areas of the rhombuses?
- A parallelogram has sides with lengths of #16 # and #15 #. If the parallelogram's area is #48 #, what is the length of its longest diagonal?
- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #3 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(3 pi)/4 #, what is the area of the parallelogram?
- A parallelogram has sides with lengths of #15 # and #12 #. If the parallelogram's area is #150 #, what is the length of its longest diagonal?

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