https://file.aiquickdraw.com/mathbot/file/17242822605860iyf1y3f.jpg

https://file.aiquickdraw.com/mathbot/file/1724282260587tkbs8cq5.jpg

To find the area of the red square in the image, we need to determine the side length of the square. 

From the image, the given measurements include the distances along the diagonal:
- One segment of the diagonal measures \(9\) units.
- The other segment measures \(6\) units.
- The entire diagonal of the square is \(15\) units.

### Step-by-Step Solution

1. **Diagonal of the Square**:
   The diagonal of the square is given as \(15\) units.

2. **Relationship between Diagonal and Side of the Square**:
   The diagonal \(d\) of a square and its side length \(s\) are related by the formula:
   \[
   d = s\sqrt{2}
   \]
   Solving for \(s\):
   \[
   s = \frac{d}{\sqrt{2}} = \frac{15}{\sqrt{2}} = \frac{15\sqrt{2}}{2}
   \]

3. **Area of the Square**:
   The area \(A\) of the square is given by:
   \[
   A = s^2 = \left(\frac{15\sqrt{2}}{2}\right)^2 = \frac{225 \times 2}{4} = \frac{450}{4} = 112.5 \text{ square units}
   \]

### Conclusion
The area of the red square is **112.5 square units**.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you derive the formula for the diagonal of a square?
2. What is the relationship between the side length and the area of a square?
3. How does the Pythagorean theorem relate to finding the diagonal of a square?
4. Can you find the side length of the square using another method?
5. How would the problem change if the square were not a perfect square but a rectangle?

**Tip:** Understanding the relationship between the diagonal and the side of a square is crucial when solving problems involving squares in geometry.

Learn how to calculate the area of a red square using the diagonal and side length. This problem involves geometry, specifically the Pythagorean theorem and formulas for square diagonals and areas. Suitable for high school students and anyone interested in geometric calculations.

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