The problem shows two squares $ABCD$ and $PQRC$ with the ratio of their sides as $3:1$. We are given the diagonal $AR$ of $\triangle AR$ as $24$ cm. The task is to find the area of square $PQRS$.

Step-by-Step Solution:

1. Side of the Larger Square $ABCD$: Let's assume the side of the larger square $ABCD$ is $3x$. Since the ratio of sides is $3:1$, the side of the smaller square $PQRC$ will be $x$.

2. Diagonal Calculation: The diagonal