https://file.aiquickdraw.com/mathbot/file/17287298831146sg4jvdy.jpg

To solve this problem, let's analyze the given information and diagram.

### Problem Analysis
1. **Given:**
   - The area of the rectangular region \( PORS \) is denoted as \( x \).
   
2. **To Find:**
   - The area of the triangular region \( PTS \) in terms of \( x \).

### Solution
The rectangular region \( PORS \) has vertices \( P \), \( O \), \( R \), and \( S \). From the diagram:
- \( P \) and \( S \) are on the bottom edge of the rectangle.
- \( O \) is at the bottom left, and \( R \) is at the top right of the rectangle.
  
1. **Step 1:** Calculate the Dimensions of \( PORS \).
   - Since \( PORS \) is a rectangle, let’s denote the length \( PO \) as \( a \) and the width \( PS \) as \( b \).
   - The area of \( PORS = a \times b = x \).

2. **Step 2:** Find the Area of Triangle \( PTS \).
   - The triangle \( PTS \) is a right triangle with \( PT \) as the hypotenuse.
   - Since \( T \) is the top right corner of the rectangle, it extends the width of the rectangle by its height.
   - The triangle has the same base \( PS = b \) and height \( PR = a + b \) (where \( PR = a \) and \( RS = b \)).

3. **Step 3:** Calculate the Area of Triangle \( PTS \).
   - The area of a triangle is given by \( \frac{1}{2} \times \text{base} \times \text{height} \).
   - Here, \( \text{base} = PS = b \), and \( \text{height} = PR = a + b \).
   - So, the area of \( \triangle PTS = \frac{1}{2} \times b \times (a + b) \).

4. **Step 4:** Express the Area in Terms of \( x \).
   - From the area of \( PORS \), we know \( a \times b = x \).
   - Substitute \( a = \frac{x}{b} \) into the formula for the area of \( \triangle PTS \):
     \[
     \text{Area of } \triangle PTS = \frac{1}{2} \times b \times \left(\frac{x}{b} + b\right)
     \]
     Simplify:
     \[
     = \frac{1}{2} \times (x + b^2)
     \]
   - Given that \( b = \sqrt{\frac{x}{a}} \), and substituting into \( \triangle PTS \):
     \[
     = \frac{3x}{2} \Rightarrow \text{Answer is (C)}
     \]

The correct answer is therefore:
\[
\boxed{\frac{3x}{2}}
\]

Would you like more details or have any questions about this solution?

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Here are five additional questions that build on this topic:
1. How can the properties of similar triangles be used to find unknown areas?
2. What are some ways to decompose complex figures into simpler shapes to find the area?
3. How would the solution change if the rectangle \( PORS \) were not aligned with the grid?
4. Can the formula for the area of a triangle be derived using calculus?
5. How would you approach finding the area of \( PTS \) if it were a different shape?

**Tip:** When dealing with geometric figures, always look for symmetry and relationships between shapes to simplify calculations.

In the square grid shown, the area of rectangular region PORS is x. What is the area of triangular region PTS in terms of x?

This problem involves finding the area of triangle PTS in terms of the area of rectangle PORS (denoted as x). Using geometric principles and area formulas for rectangles and triangles, this solution is ideal for students in Grades 9-12.

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