Square

$PR$ is the diagonal of the square $PQRS$. So we can find the sidelength of this square and it will be in the same ratio as the diagonal of the square $ABCD$.

$\triangle DOS\sim\triangle DAP$ Therefore, 

$\frac{DO}{DA}=\frac{DS}{DP}\\ \Rightarrow DP=2DS$
$DP-SP=DS\\\Rightarrow SP=DS$
$SP$ is the side of the triangle and lets name it $x$. Due to symmetry, $AP=x$ so, 

$AP^2+(DP)^2=DA^2\\ 5x^2=144\\ x=\frac{12}{\sqrt5}$
 

thus the ratio of diagonals = ratio of sides as all sqaure are similar

$\Large\frac{PR}{AC}=\frac{x}{DA}\\ \boxed{PR=\frac{12\sqrt2}{\sqrt5}}$