# A parallelogram has sides with lengths of #9 # and #8 #. If the parallelogram's area is #32 #, what is the length of its longest diagonal?

Longest diagonal

Length of longest diagonal is given by

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To find the length of the longest diagonal of a parallelogram given its side lengths and area, you can use the formula for the area of a parallelogram:

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

In a parallelogram, any diagonal can be considered as the height. Therefore, if the base is 9 and the area is 32, you can solve for the height:

[ \text{Height} = \frac{\text{Area}}{\text{Base}} = \frac{32}{9} ]

Using the Pythagorean theorem, you can find the length of the longest diagonal by considering the side lengths 8, 9, and the height as the other side:

[ \text{Longest diagonal}^2 = \text{height}^2 + \text{side length}^2 ]

[ \text{Longest diagonal}^2 = \left(\frac{32}{9}\right)^2 + 8^2 ]

[ \text{Longest diagonal}^2 = \frac{1024}{81} + 64 ]

[ \text{Longest diagonal}^2 = \frac{1024 + 5184}{81} ]

[ \text{Longest diagonal}^2 = \frac{6208}{81} ]

[ \text{Longest diagonal} = \sqrt{\frac{6208}{81}} ]

[ \text{Longest diagonal} \approx \sqrt{76.6914} ]

[ \text{Longest diagonal} \approx 8.755 ]

So, the length of the longest diagonal of the parallelogram is approximately 8.755.

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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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