Question

A parallelogram has sides with lengths of `14` and `6`. If the parallelogram's area is `28`, what is the length of its longest diagonal?

Answer 1

Longest diagonal: `2sqrt(58+2sqrt(2))`

Explanation:

In the above diagram:
`triangle PSX=triangle QRZ`
`color(white)("XXX")rarr "Area"(square PQZX)="Area"(square PQRS)`

Given
`color(white)("XXX")"Area"(square PQRS)= 28`

Let `abs(RY)=w`
then
`color(white)("XXX")wxx14=28 rarr w=2`

and by the Pythagorean Theorem
`color(white)("XXX")abs(QY)=sqrt(6^2-2^2)=sqrt(32)=4sqrt(2)`

Therefore
`color(white)("XXX")abs(PY)=14+4sqrt(2)`
and
`color(white)("XXX")abs(PY)=14+4sqrt(2)`

Applying the Pythagorean Theorem again
`color(white)("XXX")abs(PR)=sqrt((14+4sqrt(2))^2+2^2)`

`color(white)("XXXXX")=2sqrt(58+2sqrt(2))`