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A parallelogram has sides with lengths of #9 # and #8 #. If the parallelogram's area is #16 #, what is the length of its longest diagonal?

Answer 1

Aaliyah Anderson

Length of its longest diagonal is #16.9#

Area of a parallelogram is given by #axxbxxsintheta#,

where #a# and #b# are two sides of a parallelogram and #theta# is the angle included between them.

As sides are #9# and #8# and area is #16# we have

#9xx8xxsintheta=16# or #sintheta=16/(9xx8)=2/9#

#costheta=sqrt(1-(2/9)^2)=sqrt(1-4/81)#

= #sqrt(77/81)=1/9sqrt77=8.775/9=0.975#

Then larger diagonal of parallelogram would be given by

#sqrt(a^2+b^2+2abcostheta)=sqrt(9^2+8^2+2xx9xx8xx0.975#

= #sqrt(81+64+144xx0.975)=sqrt(145+140.4)#

= #sqrt285.4=16.9#

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Answer 2

Savannah Aguilar

The length of the longest diagonal of the parallelogram is ( \sqrt{9^2 + (8 - x)^2} ), where ( x ) is the length of the shorter diagonal.

Given that the area of the parallelogram is ( 16 ), we have the equation ( 9x = 16 ). Solving for ( x ), we get ( x = \frac{16}{9} ).

Thus, the length of the longest diagonal is ( \sqrt{9^2 + \left(8 - \frac{16}{9}\right)^2} ).

Simplify this to get the length of the longest diagonal.

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