A parallelogram has sides with lengths of #14 # and #9 #. If the parallelogram's area is #42 #, what is the length of its longest diagonal?

Answer 1

Here is a diagram.

The image shows that the height of this parallelogram is unknown. The formula for area of a parallelogram is #a = b xx h#. We know the base, and we know the area, so we can solve for height.

#a = b xx h#

#42 = 14 xx h#

#42/14 = h#

#3 = h#

#:.# The height of the parallelogram is 3 units.

Now that we know two sides, we can use SOHCAHTOA to determine the measure of angle B.

We know the side opposite B and the hypotenuse. We will therefore use #sin#.

#sinB = 3/9#

#B = 19˚#

Since opposite angles in parallelograms have equal measures, we can conclude that #B = D = 19˚, "or" B + D = 38˚#.

The total angles in a parallelogram add up to #360˚#, so we can state that

#A + C = 360 - 38, A = C#

Solving the system of equations:

#A + A = 322#

#2A = 322#

#A = 161˚#

Now that we know the measure of A, the length of #AB# and the length of #AD#, we can use cosine's law to determine the length of #BD#.

#BD^2 = AD^2 + AB^2 - 2ADAB(cosA)#

#BD^2 = 9^2 + 14^2 - 2(9)(14)(cos(161˚))#

#BD ~~ 22.70 #

#:.# The length of the longest diagonal is approximately #22.70# units.

Hopefully this helps!

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Answer 2

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 the base and the height. In this case, the base is the length of one of the sides, and the height is the perpendicular distance between the side and its opposite side.

Given that the area of the parallelogram is 42 and one of its sides has a length of 14, we can find the height of the parallelogram using the formula:

[ \text{Area} = \text{Base} \times \text{Height} ]

[ 42 = 14 \times \text{Height} ]

[ \text{Height} = \frac{42}{14} ]

[ \text{Height} = 3 ]

Now, since the opposite sides of a parallelogram are equal in length, the other side of the parallelogram also has a length of 14. Using the Pythagorean theorem, we can find the length of the longest diagonal, which is the hypotenuse of a right triangle formed by the two sides and the height.

[ \text{Longest diagonal}^2 = \text{Base}^2 + \text{Height}^2 ]

[ \text{Longest diagonal}^2 = 14^2 + 3^2 ]

[ \text{Longest diagonal}^2 = 196 + 9 ]

[ \text{Longest diagonal}^2 = 205 ]

[ \text{Longest diagonal} = \sqrt{205} ]

[ \text{Longest diagonal} \approx 14.32 ]

Therefore, the length of the longest diagonal of the parallelogram is approximately 14.32 units.

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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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