# A parallelogram has sides with lengths of #9 # and #8 #. If the parallelogram's area is #48 #, what is the length of its longest diagonal?

Parallelogram Area

In this case,

Pythagorean theorem :

Longer diagonal of the parallelogram =

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To find the length of the longest diagonal of a parallelogram given the lengths of its sides and its area, follow these steps:

- Use the formula for the area of a parallelogram: ( \text{Area} = \text{base} \times \text{height} ).
- Determine the height of the parallelogram using the given area and one of the side lengths.
- Use the Pythagorean theorem to find the length of the diagonal, as it forms a right triangle with the two sides of the parallelogram.

Given:

- Side lengths of the parallelogram: 9 and 8.
- Area of the parallelogram: 48.

Step 1: [ \text{Area} = \text{base} \times \text{height} ] [ 48 = 9 \times \text{height} ] [ \text{height} = \frac{48}{9} = 16/3 ]

Step 2: Using the area formula, we found the height of the parallelogram.

Step 3: In a parallelogram, the diagonals bisect each other and form two congruent right triangles. Let ( a ) and ( b ) be the side lengths of the parallelogram, and ( d ) be the length of the diagonal. Then, using the Pythagorean theorem: [ d^2 = a^2 + b^2 ]

Substitute the given side lengths: [ d^2 = 9^2 + (16/3)^2 ] [ d^2 = 81 + \frac{256}{9} ] [ d^2 = \frac{729 + 256}{9} ] [ d^2 = \frac{985}{9} ]

Take the square root of both sides to find ( d ): [ d = \sqrt{\frac{985}{9}} ] [ d \approx 10.5 ]

So, the length of the longest diagonal of the parallelogram is approximately 10.5 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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