Question

A parallelogram has sides with lengths of `16` and `9`. If the parallelogram's area is `144`, what is the length of its longest diagonal?

Answer 1

`sqrt 337`

Explanation:

`ABCD ; |AB| = b = 16, |BC| = x = 9, angle BAD = theta`
`DH` is height perp to `AB` ; `|DH| = h`
`bh = 144 Rightarrow 16h = 144 Rightarrow h = 9`
`sin theta = h/x = 9/9 = 1 Rightarrow theta = pi/2`
It's a rectangle `16 times 9`.
`ABC` is a rectangle-triangle. `BCD` too.
`|AC| = |BD| = c,` the hypotenuse.
Pythagoras say that
`c^2 = b^2 + h^2 = 16^2 + 9^2 = 256 + 81`
`c = sqrt 337`