# A parallelogram has sides with lengths of #14 # and #12 #. If the parallelogram's area is #84 #, what is the length of its longest diagonal?

Length of its longest diagonal is

Then larger diagonal of parallelogram would be given by

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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:

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

Given that the area of the parallelogram is 84 and one of its sides (considered as the base) has a length of 14, we can solve for the height:

[ 84 = 14 \times \text{height} ]

[ \text{height} = \frac{84}{14} ]

[ \text{height} = 6 ]

Now, the height of the parallelogram represents the perpendicular distance between the base and the opposite side. Since a parallelogram has opposite sides equal in length, the height also represents the length of the side perpendicular to the given base.

Let's denote this perpendicular side as ( b ). Since the height is 6, the length of this side is also 6.

Now, to find the length of the longest diagonal (denoted as ( d )), we can use the Pythagorean theorem applied to one of the right triangles formed by the diagonal, the base, and the perpendicular side:

[ d^2 = 14^2 + 6^2 ]

[ d^2 = 196 + 36 ]

[ d^2 = 232 ]

[ d = \sqrt{232} ]

[ d \approx 15.23 ]

So, the length of the longest diagonal of the parallelogram is approximately 15.23 units.

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