To solve this problem, we are asked to find the area of square $PQRS$, using the given ratio of the sides of squares $ABCD$ and $PQRC$, along with the information that $\overline{AR} = 24 \, \text{cm}$.

Given Information:

1. The ratio of the sides of squares $ABCD$ and $PQRC$ is $3:1$.
2. The length of $\overline{AR} = 24 \, \text{cm}$.

Analysis:

• Let the side length of square $PQRC$ be $x$.
• Since the ratio of the sides of $ABCD$ and $PQRC$ is $3:1$, the side length of square $ABCD$ will be $3x$.
• The point $R$ divides $\overline{AR}$ into a distance that matches the side of the larger square minus the smaller square, meaning $\overline{AR} = 3x - x = 2x$.
• From the given $\overline{AR} = 24 \, \text{cm}$, we set up the equation: $2x = 24 \, \text{cm}$ Solving for $x$: $x = \frac{24}{2} = 12 \, \text{cm}$ Thus, the side length of square $PQRC$ is $12 \, \text{cm}$, and the side length of square $ABCD$ is $3 \times 12 = 36 \, \text{cm}$.

Area of $PQRC$:

The area of square $PQRC$ is: $\text{Area of } PQRC = x^2 = 12^2 = 144 \, \text{cm}^2$

Conclusion:

The area of the smaller square $PQRS$ (the one we need to find) is a part of square $PQRC$. Based on the problem structure, this should be directly related to the provided answer choices. However, confirming the selection would require further geometry on how $S$ is placed within the square.

Would you like more detailed clarification or further analysis on the geometry of this configuration? Here are some related questions to consider:

1. What is the significance of diagonal lines like $AC$ and $PR$ in the geometry of the squares?
2. How do we compute the area of a rectangle or triangle formed within squares like these?
3. What happens if the ratio between two figures changes?
4. Can we determine the perimeter from the side length of a square?
5. How does the ratio of areas change if only side lengths are known?

Tip: When solving geometry problems involving ratios, always remember to scale all dimensions accordingly to keep track of proportions accurately.