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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.

In the adjoining figure, the ratio of the sides of the two squares ABCD and PQRC is 3:1. Also, AR = 24 cm. Find the area of PQRS.

Learn how to calculate the area of a smaller square (PQRS) based on the ratio of sides of two squares and the length of a segment in the diagram. This problem involves understanding the relationship between side lengths and area through geometry and algebra.

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