A parallelogram has sides with lengths of #24 # and #9 #. If the parallelogram's area is #18 #, what is the length of its longest diagonal?

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Answer 1 · 2017-01-14T16:52:19

32.98, nearly

Let PQRS be the parallelogram, with PQ=9 and QR = 24. Let the

altitude RN from R upon PG be h.

Then, the area = 9h=18. So, h = 2.

From the #triangleQNR, sin angleNQR#=2/24=1/12.

The longer diagonal is PR and

#angle GPS=angle PQR=pi-sin^(-1)(1/12)=175.22^o#, nearly.

Now, from #triangle PSQ#,

#PR=sqrt((PQ)^2+(QR)^2-2(PQ)(QR)cosanglePQR)#

#=sqrt(9^2+24^2+2(9)(24)sqrt(1-1/12^2))#

#=sqrt(657+39sqrt143)#

=32.98, nearly.