Question

A parallelogram has sides with lengths of `24` and `9`. If the parallelogram's area is `54`, what is the length of its longest diagonal?

Answer 1

`D=32,8`

Explanation:

The area is `Area=24*9*sintheta=54` we can deduce `sintheta=54/(24*9)=1/4`
we have to solution to this goniometric equation
`theta_1=arcsin(1/4)~~14,48°` and `theta_2=pi-arcsin(1/4)~~165,52°` the first opposite the shortest diagonal and the second opposite the longest diagonal whise length is `D=root2(24^2+9^2-2*24*9*costheta_2)` or equivalently `D=root2(24^2+9^2+2*24*9*costheta_1)`
`D=root2(576+81+418,28)~~32,8`