rectangular painting

If we call the width $w$ , we can express the length as 2 $w$ +5. We get the equation, concerning the diagonal:

$w^{2}$ + $(2w+5)^{2}$ = $(5\sqrt{97} )^{2}$
$5w^{2}+20w+25=2425$
$w^{2}+4w+5=485$ $w^{2}+4w-480=0$ .

We can now solve this using the quadratic formula to get $w$ =20 and therefore length=45. The perimeter is $20 \times 2 + 45 \times 2$ = 130 .