A parallelogram has sides with lengths of #15 # and #12 #. If the parallelogram's area is #120 #, what is the length of its longest diagonal?

2 Answers
Veddesh #phi#
Mar 27, 2016

There is no formula for the area of parallelogram with its diagonals.So, there are no answers for this

P dilip_k
Mar 27, 2016

#=25.24#

Explanation:

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Let in the parallelogram ABCD the angle between two sides #AB=15 and BC= 12# be # /_ABC=theta#
Area of the parallelogram #=AB*BC*sintheta =120 =>15*12sintheta =120=>sintheta=2/3#
#:. theta =sin^-1(2/3)=41.8 deg#
So diagonal BD will be longest which is opposite to obtuse angle #/_BCD#
Now #BD^2=BC^2+CD^2-2BC*CDcos/_BCD#
#=>BD^2=BC^2+CD^2-2BC*CDcos(180-/_ABC)#
#=>BD^2=BC^2+CD^2-2BC*CDcos(180-theta)#
#=>BD^2=BC^2+CD^2+2BC*CDcos(theta)#

#=>BD^2=12^2+15^2+2*12*15*sqrt(1-sin^2theta#
#=>BD^2=12^2+15^2+2*12*15*sqrt(1-(2/3)^2#
#=>BD^2=12^2+15^2+2*12*cancel15^5*sqrt5/cancel3#
#=>BD^2=144+225+268.33#

#=>BD=sqrt(144+225+268.33)=25.24#

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