Question

A parallelogram has sides with lengths of `24` and `9`. If the parallelogram's area is `66`, what is the length of its longest diagonal?

Answer 1

The length of the longer diagonal `d=32.6855" "`units

Explanation:

The required in the problem is to find the longer diagonal `d` given Area of the parallelogram `A=66`

Area of the parallelogram `A=base * height=b*h`
Let base `b=24`
Let other side `a=9`
Let the height `h=A/b`

Solve for height `h`
`h=A/b=66/24`

`h=11/4`

Let `theta` be the larger interior angle which is opposite the longer diagonal `d`.

`theta=180^@-sin^-1 (h/a)=180^@-17.7916^@`
`theta=162.208^@`

By the Cosine Law, we can solve now for `d`

`d=sqrt((a^2+b^2-2*a*b*cos theta))`
`d=sqrt((9^2+24^2-2*9*24*cos 162.208^@))`
`d=32.6855" "`units

God bless....I hope the explanation is useful.