# A parallelogram has sides with lengths of #16 # and #9 #. If the parallelogram's area is #120 #, what is the length of its longest diagonal?

Longest diagonal measures 22.2755

Given

Area of the parallelogram = l * h = 120

AE = DF = a = sqrt(w^2-h^2) = sqrt(9^2 - 7.5^2) = 4.9749#

AF = l + a = 16 + 4.9749 = 20.9749#

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 base times height. In this case, the base could be either of the sides, and the height would be the length of the other side. However, we need to determine which side is the base and which is the height.

Let's denote the sides of the parallelogram as ( a = 16 ) and ( b = 9 ). We know that the area of the parallelogram is 120. So, using the formula for the area ( A = b \times h ), where ( b ) is the base and ( h ) is the height, we can solve for the height:

[ 120 = 16 \times h ]

Solving for ( h ), we get:

[ h = \frac{120}{16} = 7.5 ]

Now, since the height of the parallelogram is perpendicular to the base, we can consider the longer side, 16, as the base, and the height as 7.5.

Using the Pythagorean theorem, we can find the length of the longest diagonal (the hypotenuse of the triangle formed by the sides and the diagonal):

[ c^2 = a^2 + b^2 ]

[ c^2 = 16^2 + 7.5^2 ]

[ c^2 = 256 + 56.25 ]

[ c^2 = 312.25 ]

[ c = \sqrt{312.25} ]

[ c \approx 17.66 ]

So, the length of the longest diagonal of the parallelogram is approximately 17.66.

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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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