# A parallelogram has sides with lengths of #15 # and #8 #. If the parallelogram's area is #96 #, what is the length of its longest diagonal?

ABCD is the parallelogram.

Parallelogram area

where b=base, h=height

Given

Note that triangle CDE is a right-angled triangle :

Again, note that triangle CAE is also a right-angled triangle:

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The length of the longest diagonal of the parallelogram can be calculated using the formula: ( \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 them. Since a parallelogram has opposite sides equal in length, we can consider (a = 15) and (b = 8). The area of the parallelogram can be calculated using the formula: ( \text{Area} = ab\sin(\theta) ). Solving for ( \sin(\theta) ) gives ( \sin(\theta) = \frac{\text{Area}}{ab} ). Given that the area is 96, we find ( \sin(\theta) = \frac{96}{15 \times 8} = \frac{4}{5} ). Then, using the law of cosines, we can find the length of the diagonal. Substituting the known values into the formula gives ( \text{Diagonal} = \sqrt{15^2 + 8^2 + 2 \times 15 \times 8 \times \frac{4}{5}} = \sqrt{225 + 64 + 192} = \sqrt{481} \approx 21.93 ). Therefore, the length of the longest diagonal of the parallelogram is approximately 21.93 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.

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- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #6 # and sides C and D have a length of # 1 #. If the angle between sides A and C is #(7 pi)/12 #, what is the area of the parallelogram?
- A parallelogram has sides with lengths of #14 # and #15 #. If the parallelogram's area is #45 #, what is the length of its longest diagonal?

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