Question

Two opposite sides of a parallelogram each have a length of `1`. If one corner of the parallelogram has an angle of `(3 pi)/4` and the parallelogram's area is `5`, how long are the other two sides?

Answer 1

`5sqrt(2)`

Explanation:

Let us call this parallelogram `ABCD`, and the unknown lengths (`AD` and `BC`) `x`. Given that one angle (say `B`) is `frac(3pi)(4)`, or `135` degrees, the supplementary angle (or `A`) in the parallelogram is `45` degrees. Then, draw down the height to `AD`. Because `angleA=45`, then the height is `frac(1)(sqrt(2))`. Now we know the height (`frac(1)(sqrt(2))`) and the base (which we declared as `x`). Since the area of a parallelogram is `A=bh`, `5=xfrac(1)(sqrt(2))`. So, the unknown length is simply `5sqrt(2)`.