A parallelogram has sides with lengths of #15 # and #12 #. If the parallelogram's area is #99 #, what is the length of its longest diagonal?

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Answer 1 · 2017-05-25T06:51:10

The length of the longest diagonal is #=25.9#

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Let #AB=15#

and #AD=BC=12#

#AC# is the longest diagonal

Then,

the area of the parallegram is

#AB*AD*sin(hat(BAD))=99#

Therefore,

#sin(hat(BAD))=99/(15*12)=0.55#

#hat(BAD)=33.37º#

Then,

the angle #hat(ABC)=180-33.37=146.63º#

We apply the cosine rule to the triangle #DeltaABC#

#AC^2=AB^2+BC^2-2*AB*BC*cos(hat(ABC))#

#=15^2+12^2-2*15*12*cos146.63º#

#=669.65#

So,

#AC=sqrt669.65=25.9#