# A parallelogram has sides with lengths of #24 # and #9 #. If the parallelogram's area is #96 #, what is the length of its longest diagonal?

Length of the longest diagonal AC is 32.3109

Given

Area of the parallelogram = l * h = 96

AE = DF = a = sqrt(w^2-h^2) = sqrt(9^2 - 4^2) = 8.0623#

AF = l + a = 24 + 8.0623 = 32.0623#

Longest diagonal AC

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To find the length of the longest diagonal of the parallelogram, we can use the formula for the area of a parallelogram, which is given by the product of its base and height. Since the parallelogram's area is given as 96, we can express this as the product of one of its sides and the corresponding height.

Let's denote the length of the base of the parallelogram as 24 and the length of the corresponding height as h. Then, we have:

[ 96 = 24 \times h ]

Solving for h:

[ h = \frac{96}{24} = 4 ]

Now, to find the length of the longest diagonal, we use the Pythagorean theorem. In a parallelogram, the diagonals bisect each other, forming two congruent right triangles.

Using the lengths of the sides of the parallelogram, the length of one diagonal (d) can be found using:

[ d^2 = 24^2 + (2h)^2 ]

[ d^2 = 24^2 + (2 \times 4)^2 ]

[ d^2 = 576 + 64 ]

[ d^2 = 640 ]

[ d = \sqrt{640} ]

[ d \approx 25.3 ]

So, the length of the longest diagonal of the parallelogram is approximately 25.3 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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