Question

A parallelogram has sides with lengths of `12` and `8`. If the parallelogram's area is `32`, what is the length of its longest diagonal?

Answer 1

`"diagonal" = 19.7`

Explanation:

In a parallelogram, the end of the longer diagonal extends beyond the end of the base, but by how much? Let's call it `x`.

The `height = "Area"/"base" rArr "h` `= 32/12" = 8/3`

Consider the right-angled triangle at the side of the parallelogram:
It has sides of `x and 8/3` and the hypotenuse is 8.

`x^2 = 8^2 - (8/3)^2 = 512/9 " " rArr x = sqrt(512/9) = sqrt512/3`

The length of the base produced is therefore `(12 + sqrt512/3)`

Working in the new right-angled triangle, the length of the longer diagonal is now the hypotenuse of the triangle with sides of
`(12 + sqrt512/3) and 8/3`

Using Pythagoras: `"diagonal"^2 = (12 +sqrt512/3)^2 + (8/3)^2`
`"diagonal"^2 = 389.019`
`"diagonal" = 19.7`

Would working with surds rather than resorting to decimals be easier?