A parallelogram has sides with lengths of #16 # and #9 #. If the parallelogram's area is #72 #, what is the length of its longest diagonal?

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Answer 1 · 2017-10-24T02:10:57

#d_"longest" ~~ 24.2 #

Let #b = 16 and a = 9#

The area of a parallelogram is:

#"Area" = bh#

The height of the parallelogram is:

#h = "Area"/b#

#h = 72/16#

#h = 4.5#

Another equation for h is:

#h = (a)sin(theta)#

where #theta# is the angle between sides a and b.

#9sin(theta) = 4.5#

#sin(theta) = 1/2#

The other angle is #pi-theta#. To find the length of the longest diagonal, we will need its cosine:

#cos(pi-theta) = cos(pi)cos(theta) + sin(pi)sin(theta)#

#cos(pi-theta) = -cos(theta)#

#-cos(theta) = -sqrt(1-sin^2(theta))#

#cos(pi-theta) = -sqrt(1-(1/2)^2)#

#cos(pi-theta) = -sqrt3/2#

Using the Law of Cosines:

#d_"longest" = sqrt(a^2+b^2-2(a)(b)cos(pi-theta))#

#d_"longest" = sqrt(9^2+16^2-2(9)(16)(-sqrt3/2))#

#d_"longest" ~~ 24.2 #