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?

Answer 1

Longest diagonal #color(blue)(d_1 = 16.55)# units

Length of longest diagonal is given by

#d_1 = sqrt (a^2 + b^2 + 2a b cos theta)#
Where a and b are the two pairs of parallel lines and #theta# the acute angle between non parallel sides
To find #theta# first.
#Area # #A_p = a b sin theta#
Given #a = 9, b = 8, A_p = 32#
#sin theta=32 / (9 * 8) = 4/9#
#theta = sin ^-1 (4/9) = 0.4606#
#Longest diagonal #d_1 = sqrt(9^2 + 8^2 + (2 * 9 * 8 * cos 0.4606))#
#color(blue)(d_1 = 16.55# units
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Answer 2

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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Answer from HIX Tutor

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.

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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