A parallelogram has sides with lengths of #15 # and #12 #. If the parallelogram's area is #90 #, what is the length of its longest diagonal?

Answer 1

Length of its longest diagonal is #26.09#

Area of a parallelogram is given by #axxbxxsintheta#,
where #a# and #b# are two sides of a parallelogram and #theta# is the angle included between them.
As sides are #15# and #12# and area is #90# we have
#15xx12xxsintheta=90# or #sintheta=90/(15xx12)=1/2#
#costheta=sqrt(1-(1/2)^2)=sqrt(1-1/4)=sqrt(3/4)=sqrt3/2#

Then larger diagonal of parallelogram would be given by

#sqrt(a^2+b^2-2abcostheta)=sqrt(15^2+12^2+2xx15xx12xxsqrt3/2#
= #sqrt(225+144+180xx1.732)=sqrt(369+311.76)#
= #sqrt680.76=26.09#
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Answer 2

The length of the longest diagonal of the parallelogram can be found using the formula:

[ \text{Longest diagonal} = \sqrt{a^2 + b^2 + 2ab \cos(\theta)} ]

where (a) and (b) are the lengths of the sides of the parallelogram and (\theta) is the angle between them.

Given (a = 15) and (b = 12), and knowing that the area of the parallelogram is given by (90), we can find the value of (\theta) using the formula for the area of a parallelogram:

[ \text{Area} = ab \sin(\theta) ]

Substituting the given values:

[ 90 = 15 \times 12 \times \sin(\theta) ]

[ \sin(\theta) = \frac{90}{180} = \frac{1}{2} ]

Thus, ( \theta = \frac{\pi}{6} ) radians.

Now, plug the values of (a), (b), and ( \theta ) into the formula for the longest diagonal:

[ \text{Longest diagonal} = \sqrt{15^2 + 12^2 + 2 \times 15 \times 12 \times \cos\left(\frac{\pi}{6}\right)} ]

[ \text{Longest diagonal} = \sqrt{225 + 144 + 360} ]

[ \text{Longest diagonal} = \sqrt{729} ]

[ \text{Longest diagonal} = 27 ]

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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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