# A parallelogram has sides with lengths of 16  and 15 . If the parallelogram's area is 8 , what is the length of its longest diagonal?

Nov 9, 2016

$d = \sqrt{481 + 16 \sqrt{899}} = 30.9957 \ldots$

#### Explanation:

Let ${h}_{a}$ be the height relative to side $a = 16$

Then
$A = {h}_{a} \cdot a$
so
$8 = {h}_{a} \cdot 16$
and
${h}_{a} = \frac{1}{2}$

Then the projection of b on a is

${p}_{b} = \sqrt{{15}^{2} - \frac{1}{2} ^ 2} = \frac{\sqrt{899}}{2}$

Finally the diagonal is

$d = \sqrt{{\left(16 + \frac{\sqrt{899}}{2}\right)}^{2} + \frac{1}{2} ^ 2} = \sqrt{481 + 16 \sqrt{899}}$