A parallelogram has sides with lengths of #15 # and #8 #. If the parallelogram's area is #27 #, what is the length of its longest diagonal?

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Answer 1 · 2017-12-05T04:44:12

#D~~ 22.866#

Let the first side, #A = 15#
Let the second side, #B = 8#

The area is the cross-product of the sides:

#"Area" = A xx B#

#"Area" = |A||B|sin(theta)#

where #theta# is the measure of the smallest angle between #A# and #B#

Substitute the values for the area, A, and B:

#27 = (15)(8)sin(theta)#

#27 = (120)sin(theta)#

#theta = sin^-1(27/120)#

Let #alpha =# the other angle # = pi-theta#

#alpha = pi - sin^-1(27/120)#

Let #D =# the longest diagonal.

We can use the Law of Cosines to compute the length of the longest diagonal:

#D^2 = A^2+B^2-2(A)(B)cos(alpha)#

#D^2 = 15^2+8^2-2(15)(8)cos(pi - sin^-1(27/120))#

#D = sqrt(15^2+8^2-2(15)(8)cos(pi - sin^-1(27/120)))#

#D~~ 22.866#