Question

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

Answer 1

Longest diagonal: `color(green)(sqrt(657+90sqrt(23)))~~color(green)(33.0)`

Explanation:

Since the area is given as `9` (sq.units), if we use a side with length `9` as the base,
the height of the parallelogram must be `color(brown)1`

Taking the extension of the base to a point perpendicularly below the opposite vertex of the parallelogram and denoting the length of this extension as `color(blue)a`,
by the Pythagorean Theorem
`color(white)("XXX")color(blue)a=sqrt(24^2-1^2) =5sqrt(23)`

Denoting the longest diagonal as `color(red)b`
and re-applying the Pythagorean Theorem using the base plus its extension (`9+color(red)a`) and the height `color(brown)1`
`color(white)("XXX")color(red)b=sqrt((9+color(blue)(5sqrt(23)))^2+color(brown)1^2)=sqrt(81+90sqrt(23)+575+1)=sqrt(657+90sqrt(23))`

We can use a calculator to derive the approximation
`color(white)("XXX")sqrt(657+90sqrt(23))~~32.99431522`